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| 15 Theory slides |
| 13 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
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.
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 |
First, the definition of a logarithm will be used to isolate the variable term. 8^(2x)=3 ⇔ 2x=log_8 3
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.
f(t) = 4000 (3/2)^(12t) ↓ 230 660 = 4000 (3/2)^(12t)
log_c a = log_b a/log_b c
Use a calculator
LHS * 2=RHS* 2
Rearrange equation
Round to nearest integer
t ≈ 20
1/b* a = a/b
Calculate quotient
(a/b)^m=a^m/b^m
a*b/c= a* b/c
Use a calculator
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
The statements will be proved one at a time.
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
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
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.
Use the Exponential Property of Inequality.
1/a=a^(- 1)
(a^m)^n=a^(m* n)
a^m/a=a^(m-1)
Write as a power
a^m*a^n=a^(m+n)
LHS+x
LHS-2
.LHS /2.<.RHS /2.
Put minus sign in front of fraction
Rearrange inequality
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.
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.
The solutions to the equation are the x-coordinates of any points of intersection of the graphs.
Substitute - 0.5 for pH and solve the logarithmic equation by graphing.
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.
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.
The biconditional statement will be proved in two parts.
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 ✓
log_2(m) + log_2(n)=log_2(mn)
log_c a = log_b a/log_b c
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
LHS-x=RHS-x
Factor out x
Use the Zero Product Property
(II): LHS+1=RHS+1
(II): .LHS /100.=.RHS /100.
x= 1/100
log_c a = log_b a/log_b c
Use a calculator
Add terms
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.
Start by rewriting the right-hand side as a single natural logarithm.
b*ln(a)=ln(a^b)
Substitute values
1^a=1
a(- b)=- a * b
Identity Property of Multiplication
a-(- b)=a+b
State solutions
(I), (II): Use a calculator
Round to 3 significant digit(s)
x ≈ - 0.618
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
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.Substitute 105 for D in the given formula and solve for x.
I_0= 10^(- 12), D= 105
.LHS /10.=.RHS /10.
x ≈ 0.03
Use a calculator
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
The statements will be proved one at a time.
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
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
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.
Rewrite the right-hand side of the inequality as a single logarithm with base 10. Then, use the Logarithmic Property of Inequality.
1= log 10
log(m) - log(n)=log(m/n)
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.
3
Take common logarithms on both sides.
log(a^m)= m*log(a)
Distribute log 2 & log 3
LHS+log 2=RHS+log 2
LHS-xlog 3=RHS-xlog 3
Factor out x
.LHS /(log 2- log 3).=.RHS /(log 2- log 3).
Use a calculator
Round to 2 decimal place(s)
x ≈ - 4.42
Add and subtract terms
Calculate power
Consider a logarithmic equation with the following form. c=log_b a Choose the appropriate term for each statement.
To choose the appropriate term to complete the statement about the logarithmic equation, let's consider arbitrary values for a and b that satisfy the conditions.
Constant | Condition | Example value |
---|---|---|
a | Between 0 and 1 | 0.5 |
b | Greater than 1 | 2 |
We can substitute these values for a and b in the given equation. c=log_b a → c=log_2 0.5 We can use the Change of Base Formula and a calculator to find the value of c.
We obtained a negative value for c. With this information, we can choose the appropriate term for the statement.
If b is greater than 1 and the value of a is between 0 and 1, then the value for c is less than 0.
Like in part A, to choose the appropriate term to complete the statement about the logarithmic equation, we will consider arbitrary values for a and b that satisfy the conditions.
Constant | Condition | Example value |
---|---|---|
a | Greater than 1 | 2 |
b | Between 0 and 1 | 0.5 |
We can substitute these values for a and b in the given equation. c=log_b a → c=log_(0.5) 2 We can use the Change of Base Formula and a calculator to find the value of c.
We obtained a negative value for c. With this information, we can choose the appropriate term for the statement.
If b is between 0 and 1 and the value of a is greater than 1, then the value of c is less than 0.
Let's substitute a=0 in the given equation. c=log_b a → c=log_b 0 Recall that logarithms are only defined for positive numbers. Therefore, the expression log_b 0 is undefined. This means that there is no solution for c.
If a=0, there is no solution for c.
As we did in part C, let's substitute a=1 in the given equation. c=log_b a → c=log_b 1 Recall that any non-zero number raised to the power of 0 equals 1. Therefore, for any non-zero value of b, we know that log_b 1=0. This means that we have infinitely many solutions for b.
If a=1, there are infinitely many solutions for b.