4. Solving Logarithmic Equations and Inequalities
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Chapter 5
4.
Solving Logarithmic Equations and Inequalities
Logarithmic equations and inequalities are mathematical expressions that involve logarithms. This lesson delves into the methods and techniques to solve these equations. One can encounter equations where variables are present in the argument of a logarithm. To tackle them, various properties of logarithms, such as the Power Property, are employed. The lesson also touches upon the graphical approach to solving, where each side of an equation is considered a function, and solutions are derived from their intersections. Additionally, the importance of understanding the domain of functions is emphasized, ensuring valid solutions. This knowledge is crucial for students and professionals alike, as it forms the foundation for more advanced mathematical topics.
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Lesson Settings & Tools
| | 15 Theory slides |
| | 13 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Solving Logarithmic Equations and Inequalities
Slide of 15
A logarithmic equation is an equation that contains logarithms. Similarly, a logarithmic inequality is an inequality that contains logarithms. This lesson will discuss how to solve logarithmic equations and inequalities.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Solving Equations With the Variable in the Exponent
If both sides of an exponential equation can be written as powers with the same base, then the equation can be solved algebraically by using the Property of Equality for Exponential Equations. What happens if the sides of an exponential equation cannot be written as powers with the same base? 2^(x-1)=3^(x+1) How could this type of equation be solved? Round the answer to two decimal places.
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Discussion
Exponential Equation
An exponential equation is an equation where variable expressions occur as exponents. As with any kind of equation, there are different types of exponential equations.
| Example Equation | |
|---|---|
| With One Variable | 2^x=32 |
| With the Same Variable on Both Sides | 2^(2x)=5* 2^x |
| With the Same Base | 4^(3x)=4^(2x+3) |
| With Unlike Bases | 3^(x+4)=81^x |
| With a Rational Base | (1/2)^x=8 |
Method
Solving an Exponential Equation Using Logarithms
An exponential equation can be solved by using the definition of a logarithm. Consider the following example. 8^(2x)=3 To solve this equation, three steps must be followed.
1
Use the Definition of a Logarithm
expand_more First, the definition of a logarithm will be used to isolate the variable term.
8^(2x)=3 ⇔ 2x=log_8 3
2
Solve the Resulting Equation
expand_more 3
Check the Solution
expand_more The solution will be checked to make sure it is not an extraneous solution. To do so, x≈ 0.264 will be substituted in the original equation.
8^(2x)=3
x ≈ 0.264
8^(2( 0.264))? ≈ 3
Multiply
Multiply
8^(0.528)? ≈ 3
UseCalc
Use a calculator
2.997999... ≈ 3 ✓
Since a true statement was obtained, x≈ 0.264 is a solution to the equation.
Alternative Solution
Both sides of the original equation are positive. Therefore, instead of using the definition of a logarithm to rewrite the equation, common logarithms can be taken on both sides. Then, the Power Property of Logarithms can be used.
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Example
Eat the Chapati Before Fungi Grows!
Emily and her friends went to the beach on a cloudy afternoon and cooked some chapati.
Since they did not smoke the chapati and did not use salt in the cooking process, Emily knows that it will not be too long before the food spoils. Food spoilage is generally caused due to the action of microorganisms like fungi. Fungal growth is exponential and given by the following exponential function. f(t) = 4000 (3/2)^(12t) Here, f(t) is the amount of fungi in the chapati t hours after it has been baked. Emily knows that they cannot eat the chapati once the amount of fungi is 230 660.
a Write an exponential equation that must be solved to find out after how many hours the amount of fungi in the chapati is 230 660.
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b Solve the equation from Part A. Round the answer to the nearest integer.
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Hint
a Does the number 230 660 correspond to amount of fungi or time?
b Divide both sides of the equation by 4000 and then use the definition of a logarithm.
Solution
a Since the number 230 660 corresponds to amount of fungi, this number will be substituted for f(t) in the given formula.
f(t) = 4000 (3/2)^(12t) ↓ 230 660 = 4000 (3/2)^(12t)
b The equation from Part A will now be solved. To do so, start by dividing both sides by 4000.
230 660 = 4000 (3/2)^(12t) ⇕ 57.665= (3/2)^(12t) Now use the definition of a logarithm. This helps to remove the variable from the exponent. 57.665= (3/2)^(12t) ⇕ log_(32) 57.665=1/2t Finally, the equation can be solved for t.
log_(32) 57.665=1/2t
â–¼
Solve for t
log_c a = log_b a/log_b c
log 57.665/log 32=1/2t
UseCalc
Use a calculator
9.999998...=1/2t
MultEqn
LHS * 2=RHS* 2
19.999996...=t
RearrangeEqn
Rearrange equation
t=19.999996...
RoundInt
Round to nearest integer
t≈ 20
Just to make sure the obtained solution is not an extraneous solution, t≈ 20 will be substituted in the equation.
230 660 = 4000 (3/2)^(12t)
t ≈ 20
230 660 ? ≈ 4000 (3/2)^(12( 20))
â–¼
Evaluate right-hand side
MoveRightFacToNumOne
1/b* a = a/b
230 660 ? ≈ 4000 (3/2)^(202)
CalcQuot
Calculate quotient
230 660 ? ≈ 4000 (3/2)^(10)
PowQuot
(a/b)^m=a^m/b^m
230 660 ? ≈ 4000 (59 049/1024)
MoveLeftFacToNum
a*b/c= a* b/c
230 660 ? ≈ 236 196 000/1024
UseCalc
Use a calculator
230 660 ≈ 230 660.15625 ✓
Since a true statement was obtained, t≈ 20 is the solution to the equation. This means that the chapati will become unsafe to eat due to fungi growth about 20 hours after cooking it.
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Discussion
Exponential Property of Inequality
An exponential inequality is an inequality that involves exponential expressions.
Let b be a positive real number different than 1. The following statements hold true.
c|c Ifb<1 & Ifb>1 [1em] b^(x_1)≥ b^(x_2) & b^(x_1)≥ b^(x_2) ⇕ & ⇕ x_1≤ x_2 & x_1≥ x_2
These facts are also valid for strict inequalities.
Proof
The statements will be proved one at a time.
b>1
If b is greater than 1, the exponential function f(x) = b^x is increasing for its entire domain.
In the graph, it can be seen that f(x_1)≥ f(x_2) if and only if x_1≥ x_2. Considering the definition of f(x), the statement is proven. b^(x_1)≥ b^(x_2) ⇔ x_1 ≥ x_2
b<1
If b is greater than 0 and less than 1, then f(x) = b^x is decreasing for its entire domain.
In the graph, it is seen that f(x_1)≥ f(x_2) if and only if x_1≤ x_2. Considering the definition of f(x), the statement is proven.
b^(x_1)≥ b^(x_2) ⇔ x_1 ≤ x_2
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Example
Solving an Exponential Inequality
Back home from the beach, Emily realized that she managed to solve an exponential equation to calculate the expiration of the chapati she and her friends cooked. However, she also realized that she has not practiced solving exponential inequalities. To help her practice, she went online to find some worksheets and found an interesting inequality.
Help Emily solve the inequality!
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Hint
Use the Exponential Property of Inequality.
Solution
To solve the inequality, all numbers involved will first be written as powers of 2. To start, 0.5 will be written as 12.
0.5^x/2<4(2)^x ⇕ ( 12)^x/2<4(2)^x
Now, all the numbers can be written as powers of 2.
( 12)^x/2<4(2)^x
1/a=a^(- 1)
(2^(- 1))^x/2<4(2)^x
PowPow
(a^m)^n=a^(m* n)
2^(- x)/2<4(2)^x
a^m/a=a^(m-1)
2^(- x-1)<4(2)^x
WritePow
Write as a power
2^(- x-1)<2^2(2)^x
MultPow
a^m*a^n=a^(m+n)
2^(- x-1)<2^(2+x)
The base of the expressions on both sides is 2, which is is greater than 1. Therefore, by the Exponential Property of Inequality, the left-hand side of the inequality is less than the right-hand side if and only if the exponent on the left-hand side is less than the exponent on the right-hand side. 2^(- x-1)<2^(2+x) ⇔ - x-1<2+x The inequality can finally be solved.
- x-1<2+x
â–¼
Solve for x
AddIneq
LHS+x
- 1<2+2x
SubIneq
LHS-2
- 3<2x
DivIneq
.LHS /2.<.RHS /2.
- 3/2
MoveNegNumToFrac
Put minus sign in front of fraction
- 3/2
RearrangeIneq
Rearrange inequality
x>- 3/2
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Discussion
Logarithmic Equation
Method
Solving Logarithmic Equations Graphically
Logarithmic equations can be tricky to solve sometimes. However, if the exact solution is not needed, an approximated solution can be found by solving the equation graphically. Consider an example logarithmic equation. ln (x-2) = 2log x There are three steps to follow to solve logarithmic equations graphically.
1
Create Two Functions From the Given Equation
expand_more Each side of the equation can be considered as a function. In this case, both sides can be written as logarithmic functions.
ln (x-2)= 2log x ⇓ lcy= ln (x-2) & (I) y= 2log x & (II)
The idea is that each function should be relatively easy to graph.
2
Graph the Functions
expand_more A table of values including both functions will be made. However, the domain of each function needs to be considered first.
c|c
Domain of & Domain of y=ln (x-2) & y=2log x [1em]
x-2>0 & ⇕ & x>0 x>2 &
With the domains in mind, the table can now be constructed.
| x | y=ln (x-2) | y=2log x | ||
|---|---|---|---|---|
| ln (x-2) | y | 2log x | y | |
| 1 | - | - | 2log 1 | 0 |
| 2 | - | - | 2log 2 | ≈ 0.6 |
| 3 | ln ( 3-2) | 0 | 2log 3 | ≈ 0.95 |
| 4 | ln ( 4-2) | ≈ 0.69 | 2log 4 | ≈ 1.2 |
| 5 | ln ( 5-2) | ≈ 1.1 | 2log 5 | ≈ 1.4 |
| 12 | ln ( 12-2) | ≈ 2.3 | 2log 12 | ≈ 2.16 |
Now both functions will be graphed on the same coordinate plane.
The number of solutions to the equation is the number of points of intersection of the graphs. These graphs have one point of intersection, so there is only one solution.
3
Identify the x-coordinates of the Points of Intersection
expand_more The solutions to the equation are the x-coordinates of any points of intersection of the graphs.
The x-coordinate of the point of intersection is about 8. Therefore, the solution to the equation is x≈ 8. This can be verified by substituting 8 for x in the given equation and checking whether a true statement is obtained.
ln (x-2) = 2log x
x ≈ 8
ln ( 8-2) ? ≈ 2log 8
1.791759... ? ≈1.806179... ✓
Since a true statement was obtained, x≈ 8 is a solution to the equation. In this case, an exact solution cannot be found graphically.
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Example
Acidic, Basic, and Neutral Solutions
In her chemistry lesson, Emily is learning about the acidity and alkalinity of substances. The acidity or alkalinity of a substance is defined by the value of pH, where x is the hydrogen ion concentration of the substance measured in moles per liter.
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Hint
Substitute - 0.5 for pH and solve the logarithmic equation by graphing.
Solution
To find the hydrogen ion concentration of a solution with a pH of - 0.5, the number must be substituted for pH in the given formula.
pH=- log x substitute - 0.5=- log x
Next, to solve the equation, each side will be considered as a function.
- 0.5= - log x ⇓ y= - 0.5 & (I) y= - log x & (II)
The first function is a horizontal line whose y-intercept is - 0.5. The second is a logarithmic function. Considering that its domain is x>0, make a table of values.
| x | - log x | y |
|---|---|---|
| 0.1 | - log 0.1 | 1 |
| 0.5 | - log 0.5 | ≈ 0.3 |
| 1 | - log 1 | 0 |
| 2 | - log 2 | ≈ - 0.3 |
| 3 | - log 3 | ≈ - 0.48 |
Now, graph the functions on the same coordinate plane.
The graphs intersect at one point. The x-coordinate of the point of intersection is the hydrogen ion concentration of the solution.
The exact value of the x-coordinate cannot be identified. However, the answer must be rounded to the nearest mole per liter, so the hydrogen concentration of the solution with a pH of - 0.5 is 3, rounded to the nearest mole per liter. The last step is to check the solution by substituting it back into the equation.
Since a true statement was obtained, x≈ 3 is a solution to the equation.
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Discussion
Property of Equality for Logarithmic Equations
Let b be a positive real number different than 1. Two logarithms with the same base b are equal if and only if their arguments are equal.
log_b x=log_b y ⇔ x=y
Since logarithms are defined for positive numbers, x and y must be positive.
Proof
The biconditional statement will be proved in two parts.
log_b x=log_b y ⇒ x=y
Let a be a real number such that a=log_b x. log_b x=log_b y → a=log_b y By using the definition of a logarithm, the logarithm equation can be written as an exponential equation. a=log_b y ⇔ b^a=y Finally, log_b x can be substituted for a and the Inverse Property of Logarithms can be used.
x=y ⇒ log_b x=log_b y
The Inverse Property of Logarithms can be used to express x as b^(log_b x). x=y ⇔ b^(log_b x)=y Now, let c be a real number such that c=log_b x. b^(log_b x)=y → b^c=y Next, the definition of a logarithm can be used to rewrite the obtained exponential equation as a logarithmic equation. b^c=y ⇔ c=log_b y Finally, by its own defitnion, c is equal to log_b x. c=log_b y ⇔ log_b x=log_b y ✓
Method
Solving Logarithmic Equations by Using the Property of Equality
If both sides of a logarithmic equation can be written as a single logarithm with the same base b, with b>0 and b≠1, then the equation can be solved by using the Property of Equality for Logarithmic Equations. Consider the following logarithmic equation. log_2 x +log_2 10 = log_4 x To solve the equation, there are four steps to follow.
1
Rewrite Both Sides as a Single Logarithm With the Same Base
expand_more To rewrite the left-hand side as a single logarithm with base 2, the Product Property of Logarithms will be used. To rewrite the right-hand side as a single logarithm with base 2, the Change of Base Formula will be used.
log_2 x +log_2 10 = log_4 x
log_2(m) + log_2(n)=log_2(mn)
log_2 10x = log_4 x
log_c a = log_b a/log_b c
log_2 10x = log_2 x/log_2 4
Next, to calculate the value of log_2 4, the definition of a logarithm will be used. Keep in mind that 2^2= 4. Definition:& b^c= a ⇔ log_b a= c Expression:& 2^2= 4 ⇔ log_2 4= 2 Knowing that log_2 4=2, we can continue writing both sides of the equation as a single logarithm with the same bse.
log_2 10x = log_2 x/log_2 4
Calculate logarithm
log_2 10x = log_2 x/2
MultEqn
LHS * 2=RHS* 2
2log_2 10x = log_2 x
m* log_2(a)=log_2(a^m)
log_2 (10x)^2 = log_2 x
PowProdII
(a b)^m=a^m b^m
log_2 100x^2 = log_2 x
Both sides of the equation have been written as a single logarithm with base 2.
2
Use the Property of Equality for Logarithmic Equations
expand_more Next, the Property of Equality for Logarithmic Equations can be used to simplify the expressions.
log_2 100x^2 = log_2 x ⇕ 100x^2=x
3
Solve the Obtained Equation
expand_more Now, the obtained equation can be solved. This equation does not involve logarithms.
100x^2=x
â–¼
Solve for x
SubEqn
LHS-x=RHS-x
100x^2-x=0
FactorOut
Factor out x
x(100x-1)=0
ZeroProdProp
Use the Zero Product Property
lcx=0 & (I) 100x-1=0 & (II)
AddEqn
(II): LHS+1=RHS+1
lx=0 100x=1
DivEqn
(II): .LHS /100.=.RHS /100.
lx=0 x= 1100
Two solutions were found, x=0 and x= 1100.
4
Check the Solutions
expand_more Finally, the solutions must be checked. It is not uncommon to find extraneous solutions. The solution x=0 will be checked first.
Logarithms are not defined for non-positive numbers. Since 0 is not positive, x=0 is an extraneous solution. This means that it is not a solution to the given equation. The solution x= 1100 will be checked now.
log_2 x +log_2 10 = log_4 x
Substitute
x= 1/100
log_2 1/100 +log_2 10? = log_4 1/100
log_c a = log_b a/log_b c
log 1100/log 2+log 10/log 2? =log 1100/log 4
UseCalc
Use a calculator
- 6.643856...+3.321928...? =- 3.321928...
AddTerms
Add terms
- 3.321928...=- 3.321928... ✓
A true statement was obtained. Therefore, x= 1100 is a solution to the given equation.
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Example
Solving Equations With Logarithms on Both Sides
Emily told her study buddy about how she used a graph to solve a logarithmic equation. Her friend is pretty competitive, so he challenged Emily to solve a logarithmic equation with logarithms on both sides but without graphing.
Help Emily show her study buddy that she has a solid understanding of logarithmic equations! Round the answer to three significant figures.
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Hint
Start by rewriting the right-hand side as a single natural logarithm.
Solution
The equation involves natural logarithms. To solve it, the first step is to rewrite the right-hand side as a single logarithm by using the Power Property of Logarithms.
ln (x+1)=2ln x
b*ln(a)=ln(a^b)
ln (x+1)=ln x^2
Since both sides are single logarithms with the same base, the Property of Equality for Logarithmic Equations can be used. ln (x+1)=ln x^2 ⇕ x+1=x^2 Notice that the resulting equation is a quadratic equation. Therefore, the values of a, b, and c need to be identified. x+1=x^2 ⇕ - 1x^2+ 1x+ 1=0 Next, a= - 1, b= 1, and c= 1 will be substituted in the Quadratic Formula.
x=- b±sqrt(b^2-4ac)/2a
SubstituteValues
Substitute values
x=- 1±sqrt(1^2-4( - 1)( 1))/2( - 1)
â–¼
Evaluate right-hand side
BaseOne
1^a=1
x=- 1±sqrt(1-4(- 1)(1))/2(- 1)
MultPosNeg
a(- b)=- a * b
x=- 1±sqrt(1-(- 4)(1))/- 2
IdPropMult
Identity Property of Multiplication
x=- 1±sqrt(1-(- 4))/- 2
SubNeg
a-(- b)=a+b
x=- 1±sqrt(5)/- 2
StateSol
State solutions
lcx_1= - 1+sqrt(5)- 2 & (I) x_2= - 1-sqrt(5)- 2 & (II)
(I), (II): Use a calculator
lx_1=- 0.618033... x_2=1.618033...
RoundSigDig
Round to 3 significant digit(s)
lx_1≈ - 0.618 x_2≈ 1.62
Finally, the solutions can be verified by substituting them into the original equation. The solution x≈ - 0.618 will be checked first.
ln (x+1)=2ln x
x ≈ - 0.618
ln ( - 0.618+1)? ≈2ln - 0.618 *
Logarithms are only defined for positive numbers, so x≈ - 0.618 is an extraneous solution. The solution x≈ 1.62 will be checked now.
ln (x+1)=2ln x
x ≈ 1.62
ln ( 1.62+1)? ≈2ln 1.62
AddTerms
Add terms
ln 2.62? ≈2ln 1.62
b*ln(a)=ln(a^b)
ln 2.62? ≈ln 1.62^2
CalcPow
Calculate power
ln 2.62? ≈ln 2.6244
UseCalc
Use a calculator
0.963174... ≈ 0.964852... ✓
A true statement was found this time. Therefore, the only solution to the equation is x≈ 1.62.
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Discussion
Solving Equations With a Logarithm on One Side
If only one side of a logarithmic equation can be written as a single logarithm with base b, with b>0 and b≠1, then the equation can be solved by using the definition of a logarithm. log_b m = n ⇔ b^n=m Since logarithms are defined only for positive numbers, m must be a positive number. Consider an example equation. 4log_2 (x-1)+6=14 To solve this equation, four steps must be followed.
1
Isolate the Logarithmic Expression on One Side
expand_more The first step is to isolate the logarithmic expression. This can be done by using inverse operations.
2
Use the Definition of a Logarithm
expand_more Once the logarithmic expression is isolated, the definition of a logarithm can be used to eliminate the logarithm.
log_2 (x-1)=2 ⇔ 2^2=x-1
3
Solve the Obtained Equation
expand_more The obtained equation can now be solved.
2^2=x-1
â–¼
Solve for x
x=5
4
Check the Solutions
expand_more Finally, to detect any extraneous solutions, the obtained value can be checked by substituting it into the original equation.
A true statement was obtained, so the solution is not extraneous.
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Example
Logarithms and the Decibels at a Rock Concert
Decibels (dB) are units of measurement that are used to describe a sound's loudness. The decibels of a sound with an intensity of x watts per square meter are given by the following formula. D=10dB log x/I_0 Here, I_0 is the intensity of the softest sound a healthy human ear can detect, the value of which is 10^(- 12) Wm^2. Emily wants to attend a rock concert, but she is concerned about how loud the music may be.
A typical rock concert decibel level is about 105dB. Calculate the watts per square meter of a typical rock concert. Round the answer to the nearest hundreds.
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Hint
Substitute 105 for D in the given formula and solve for x.
Solution
To calculate the watts per square meter of a typical rock concert, 105 will be substituted for D in the given equation. The logarithmic equation will then be solved algebraically. Recall that I_0=10^(- 12) Wm^2. For simplicity, the units will not be considered until the end.
D=10log x/I_0
SubstituteII
I_0= 10^(- 12), D= 105
105=10log x/10^(- 12)
DivEqn
.LHS /10.=.RHS /10.
10.5=log x/10^(- 12)
Next, to eliminate the logarithm, its definition will be used. Since the base is not stated, the formula involves a common logarithm. 10.5=log x/10^(- 12) ⇕ 10^(10.5)=x/10^(- 12) Finally, the equation can be solved for x.
10^(10.5)=x/10^(- 12)
x≈ 0.03
A rock concert with a decibel level is 105dB has a sound with an intensity of 0.03 watts per square meter. To make sure that this is not an extraneous solution, it can be confirmed by substituting the value back into the equation.
105=10log x/10^(- 12)
x ≈ 0.03
105? ≈10log 0.03/10^(- 12)
UseCalc
Use a calculator
105≈ 104.771212... ✓
Since a true statement was obtained, x≈ 0.03 is a solution to the equation.
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Discussion
Logarithmic Property of Inequality
A logarithmic inequality is an inequality that involves logarithms.
Let b be a positive real number different than 1. The following statements hold true.
c|c Ifb<1 & Ifb>1 [1em] log_b x_1≥ log_b x_2 & log_b x_1≥ log_b x_2 ⇕ & ⇕ x_1≤ x_2 & x_1≥ x_2
These facts are also valid for strict inequalities.
Proof
The statements will be proved one at a time.
b>1
If b is greater than 1, the logarithmic function f(x) = log_b x is increasing over its entire domain.
In the graph, it can be seen that f(x_1)≥ f(x_2) if and only if x_1≥ x_2. Considering the definition of f(x), the statement is proven. log_b x_1≥ log_b x_2 ⇔ x_1≥ x_2
b<1
If b is greater than 0 and less than 1, then f(x) = log_b x is decreasing over its entire domain.
In the graph, it can be seen that f(x_1)≥ f(x_2) if and only if x_1≤ x_2. Considering the definition of f(x), the statement is proven.
log_b x_1≥ log_b x_2 ⇔ x_1≤ x_2
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Example
Solving a Logarithmic Inequality
After going to the rock concert and using logarithms to calculate the watts per square meter, Emily wants to finish this topic on a high note. Her teacher asked her to solve a logarithmic inequality for extra credit.
Write the answer in interval notation.
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Hint
Rewrite the right-hand side of the inequality as a single logarithm with base 10. Then, use the Logarithmic Property of Inequality.
Solution
To solve the inequality, the first step is writing the right-hand side as a single logarithm with base 10. Start by writing 1 as a logarithmic expression by using the definition of a logarithm.
10^1=10 ⇔ 1=log 10
With this information, log 10 can be substituted for 1. Then, the Quotient Property of Logarithms can be used.
log (x-3)
Substitute
1= log 10
log (x-3)
log(m) - log(n)=log(m/n)
log (x-3)
Both sides are written as single logarithms with base 10, which is greater than 1. Therefore, by the Logarithmic Property of Inequality, the left-hand side of the inequality is less than the right-hand side if an only if the argument of the left-hand side is less than the argument of the right-hand side.
log (x-3)
x-3 < x+6/10
x<4
Finally, recall that logarithms are only defined for positive numbers. Therefore, the arguments of the logarithms in the original inequality must be greater than zero.
| log (x-3) | |
|---|---|
| Argument on the LHS | Argument on the RHS |
| x-3>0 ⇕ x>3 |
x+6>0 ⇕ x>- 6 |
The solution set is the intersection of three inequalities.
Inequality I:& x<4 Inequality II:& 3< x Inequality III:& - 6
The solution to the inequality are all real numbers greater than 3 and less than 4 because the three lines overlap between 3 and 4. This solution can be written as a compound inequality or in interval notation.
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Closure
Exponential Expressions With Different Bases
With the topics covered in this lesson, the challenge presented at the beginning can now be solved. Consider the given exponential equation again. 2^(x-1)=3^(x+1) Here, each side is an exponential expression. What is the value of x? Round the answer to two decimal places.
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Hint
Take common logarithms on both sides.
Solution
Since the expressions have different bases, the Property of Equality for Exponential Equations cannot be used. However, both 2^(x-1) and 3^(x+1) are greater than zero for any value of x. Therefore, common logarithms can be taken on both sides.
2^(x-1)=3^(x+1) ⇕ log 2^(x-1) = log 3^(x+1)
Now use the Power Property of Logarithms. Then the variable x can be isolated.
log 2^(x-1) = log 3^(x+1)
log(a^m)= m*log(a)
(x-1)log 2 = (x+1)log 3
Distr
Distribute log 2 & log 3
xlog 2-log 2 = xlog 3+log 3
AddEqn
LHS+log 2=RHS+log 2
xlog 2= xlog 3+log 3+log 2
SubEqn
LHS-xlog 3=RHS-xlog 3
xlog 2- xlog 3=log 3+log 2
FactorOut
Factor out x
x(log 2- log 3)=log 3+log 2
DivEqn
.LHS /(log 2- log 3).=.RHS /(log 2- log 3).
x=log 3+log 2/log 2- log 3
UseCalc
Use a calculator
x=- 4.419022...
RoundDec
Round to 2 decimal place(s)
x≈ - 4.42
The solution to the exponential equation is x≈ - 4.42. This can verified by substituting it into the original equation.
2^(x-1)=3^(x+1)
x ≈ - 4.42
2^(- 4.42-1)? ≈3^(- 4.42+1)
AddSubTerms
Add and subtract terms
2^(- 5.42)? ≈3^(- 3.42)
CalcPow
Calculate power
0.023357... ≈ 0.023347... ✓
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Solving Logarithmic Equations and Inequalities
Exercise 1.1
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Let's start by dividing both sides of the equation by 2. 2(3)^x=10 ⇔ 3^x=5 Now we can use the definition of a logarithm to rewrite the exponential equation as a logarithmic equation. Definition:& b^c= a ⇔ log_b a= c Equation:& 3^x= 5 ⇔ log_3 5= x Finally, we will use the Change of Base Formula to find the numerical value of log_3 5. Because the calculator can only calculate logarithms with base 10 or base e, we will use common logarithms, which have base 10. Let's not forget to round the answer to three significant digits!
Let's start by dividing both sides of the equation by 5. 5(1/2)^(x+1)=2 ⇔ (1/2)^(x+1)=2/5 Now we can use the definition of a logarithm to rewrite the exponential equation as a logarithmic equation. Definition b^c= a ⇔ log_b a= c [1em] Equation ( 1/2)^(x+1)= 2/5 ⇔ log_(12) 2/5= x+1 Finally, we will use the Change of Base Formula to find the numerical value of log_(12) 25. As in Part A, because the calculator can only calculate logarithms with base 10 or base e, we will use common logarithms, which have base 10. Let's not forget to round the answer to three significant digits!
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Let's start by writing both sides of the inequality as a single power of 3. To do this, we will first rewrite 27 as 3^3. Then, we will use the Product of Powers Property and the Quotient of Powers Property. Let's do it!
Now, we can use the Exponential Property of Inequality. Note that the base of the powers is 3, which is greater than 1. Therefore, the expression on the left-hand side is greater than or equal to the expression on the right-hand side if and only if the exponent on the left-hand side is greater than or equal to the exponent on the right-hand side. 3^(3+2x) ≥ 3^(x-1) ⇕ 3+2x≥ x-1 We can now solve the obtained linear inequality for x.
Let's start by writing both sides of the inequality as a single power of 4. To do this, we will first rewrite 116 as 4^(- 2) and 64 as 4^3. Then, we will use the Product of Powers Property and the Power of a Power Property. Let's do it!
Now, we can use the Exponential Property of Inequality. Note that the base of the powers is 4, which is greater than 1. Therefore, similar to Part A, the expression on the left-hand side is less than the expression on the right-hand side if and only if the exponent on the left-hand side is less than the exponent on the right-hand side. 4^(- 2+x) < 4^(3x) ⇕ - 2+x< 3x Let's finally solve the linear inequality for x.
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Solution
Solve the logarithmic equation graphically. - log x+3=ln (x-1.5) Round the answer to the nearest integer.
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To solve the logarithmic equation graphically, we will start by considering each side of the equation as a logarithmic function. - log x+3= ln (x-1.5) ⇓ y=- log x+3 and y=ln (x-1.5) Recall that the argument of a logarithm must be positive. With this information, we can write the domain of each function.
| Function | Positive argument | Domain |
|---|---|---|
| y=- log x+3 | x>0 | x>0 |
| y=ln (x-1.5) | x-1.5>0 | x>1.5 |
Keeping in mind the domain of each function, we can make a table of values to find points on each curve. To do this, we need to use a calculator to calculate common and natural logarithms.
| y=- log x+3 | y=ln (x-1.5) | |||
|---|---|---|---|---|
| x | - log x+3 | y | ln (x-1.5) | y |
| 0.5 | - log 0.5+3 | ≈ 3.3 | - | - |
| 1 | - log 1+3 | 3 | - | - |
| 2 | - log 2+3 | ≈ 2.7 | ln( 2-1.5) | ≈ - 0.7 |
| 3 | - log 3+3 | ≈ 2.5 | ln( 3-1.5) | ≈ 0.4 |
| 4 | - log 4+3 | ≈ 2.4 | ln( 4-1.5) | ≈ 0.9 |
| 5 | - log 5+3 | ≈ 2.3 | ln( 5-1.5) | ≈ 1.3 |
| 10 | - log 10+3 | 2 | ln( 10-1.5) | ≈ 2.1 |
Let's now plot on a coordinate plane the points that we obtained in the table. We will also connect them with smooth curves.
The x-coordinate of the point of intersection is the solution to our equation.
Although we cannot determine an exact answer, we see that, rounded to the nearest integer, the solution is x≈ 9.
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The left-hand side of the logarithmic equation is already expressed as a single logarithm. To express the right-hand side as a single logarithm, we can use the Power Property of Logarithms. Then, we will use the Property of Equality for Logarithmic Equations to find the value of x. Let's do it!
We now have a quadratic equation. Let's identify the values of a, b, and c. x^2+x-2=0 ⇕ 1x^2+ 1x+( - 2)=0 We can find the value of x by substituting these values into the Quadratic Formula.
Finally, we will see if any of the obtained solutions are an extraneous solution. To do so, we will substitute x_1=1 and x_2=- 2 in the given equation. Let's start with x_1=1.
We obtained a true statement. This means that x_1=1 is a solution to the equation. Let's check x_2=- 2 by following the same procedure.
Note that in the right-hand side of the equation we have a logarithm with a negative argument. Since the argument of a logarithm must always be non-negative, x_2=- 2 is an extraneous solution. Therefore, the given equation has only one solution which is x=1.
This time, the right-hand side of the equation is expressed as a single logarithm. However, we need to rewrite the left-hand side as a single logarithm. To do this, we will use the Power Property of Logarithms. Keep in mind that (sqrt(x))^3 is equal to x.
Now that both sides are expressed as single logarithms, we can use the Property of Equality for Logarithmic Equations, and solve for x.
Finally, let's see if x=4 is an extraneous solution.
We obtained a true statement. Therefore, x=4 is a solution to the given equation.
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Solution
Solve the logarithmic equation. 3log_5 (2x+1)=15
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To use the definition of a logarithm, we need to isolate the logarithmic expression on one side. To do this, we will divide both sides of the equation by 3. 3log_5 (2x+1)=15 ⇕ log_5 (2x+1)=5 Now, to continue solving the logarithmic equation, we can use the definition of a logarithm. Definition:& log_b a= c ⇔ b^c= a Equation:& log_5 ( 2x+1)= 5 ⇔ 5^5= 2x+1 We obtained a linear equation. Let's solve it for x.
Let's check if x=1562 is an extraneous solution.
Since we obtained a true statement, x=1562 is a solution to the equation.
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Solve the logarithmic inequality. log_2 (x-4) < log_2(2x+8) - 2 Write the answer in interval notation.
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We want to solve an inequality. To do so, we will start by writing 2 as a logarithm with base 2. Let's use the definition of a logarithm. 2^2= 4 ⇔ log_2 4= 2 Therefore, in the given inequality, we can substitute log_2 4 for 2. log_2 (x-4) < log_2(2x+8) - 2 ⇕ log_2 (x-4) < log_2(2x+8) - log_2 4 Next, we can use the Quotient Property of Logarithms to express the right-hand side as a single logarithm. log_2 (x-4) < log_2(2x+8) - log_2 4 ⇕ log_2 (x-4) < log_2 2x+8/4 Now both sides are expressed as a single logarithm with the same base. Therefore, by the Logarithmic Property of Inequality, we can eliminate the logarithms. Note that the base of the logarithms is 2, which is greater than 1. log_2 (x-4) < log_2 2x+8/4 ⇕ x-4< 2x+8/4 Let's solve the obtained linear inequality.
Finally, recall that logarithms are only defined for positive numbers. Therefore, the arguments of the logarithms in the original inequality must be greater than zero.
| log_2 (x-4) < log_2(2x+8) - 2 | |
|---|---|
| Argument on the LHS | Argument on the RHS |
| x-4>0 ⇕ x>4 |
2x+8>0 ⇕ x>- 4 |
The solution set is the intersection of three inequalities.
Inequality I:& x<12
Inequality II:& 4< x
Inequality III:& - 4
The solution to the inequality are all real numbers greater than 4 and less than 12. This solution can be written as a compound inequality or in interval notation.
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Solution
Solve the exponential equation. 4(2)^(2x)=3^x/9 Round the answer to one decimal place.
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To solve the exponential equation, we will start by writing both sides as single powers. To do so, we will rewrite 4 as 2^2 and 9 as 3^2, and use the Product of Powers Property and the Quotient of Powers Property.
Now that both sides are expressed as single powers, we can take logarithms on both sides. Then, to remove the variable from the exponents, we will use the the Power Property of Logarithms.
Finally, we can solve the obtained linear equation for x. Let's. do it!
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Solving Logarithmic Equations and Inequalities
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