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1. Properties of Logarithms
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Chapter 5
1. 

Properties of Logarithms

This lesson delves into the intriguing world of logarithms, highlighting their inverse relationship with exponents. Through various examples and explanations, readers are introduced to the properties of logarithms and their applications. Paulina, a character in the content, embarks on a journey to understand and master logarithmic expressions. She learns to rewrite exponential equations as logarithmic ones and vice versa. As she progresses, Paulina discovers how to evaluate, expand, and condense logarithmic expressions, both algebraic and numeric. By the end, she feels confident in her understanding and application of logarithmic properties.
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11 Exercises - Grade E - A
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Properties of Logarithms
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Inverse operations are mathematical operations that, in a way, undo each other. This lesson will discuss logarithims, which are the inverse operation of raising a number to a variable. Properties of logarithms will also be described.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

What Is the Inverse Operation of Calculating an Exponent?

Sometimes the inverse of a mathematical operation is clearly identifiable. For example, addition and subtraction, multiplication and division, or raising to the power of n and calculating the n^\text{th} root are inverse operations.
Inverse Operations: 1) Addition and Subtraction 2) Multiplication and Division 3) Power and Root
However, in some other cases, inverse operations are a bit more difficult to come up with. What is an operation that undoes calculating an exponent?
inverse of an exponent
Discussion

Logarithm

A logarithm is the inverse function of an exponential function. The logarithm of a positive number m is written as log_b m and read as the logarithm of m with base b.


log_b m=n ⇔ b^n=m

Here, b is called the base in both the logarithm and the exponential expression. Logarithms are defined only for positive values of b and m, where b is not equal to 1. To see the implications of this definition, a particular example will be considered. log_4 16=n In this equation, the definition of logarithm implies that n is the exponent to which the base 4 must be raised in order to obtain 16. log_4 16= n & ⇔ 4^n=16 The following diagram illustrates how a logarithmic form has an equivalent exponential form using example numerical values.
Equivalence between logarithmic and exponential expressions

Why

 Logarithms are only defined for positive values

Logarithms are undefined for non-positive values. The reason behind this can be explained by rewriting the logarithm in exponential form. y=log_b x ⇔ b^y=x Because b is a positive number, the expression b^y is always positive. Therefore, since b^y=x, the value of x is always positive. This means that x can never be negative. In this case, x is the expression in the logarithm. Therefore, expressions in logarithmic functions must be positive.

As a consequence of the definition of a logarithm, two properties can be deduced. In these properties, b is positive and not equal to 1.

Property Reason
log_b b=1 A number raised to the power of 1 is equal to itself.
log_b 1=0 A number raised to the power of 0 is equal to 1.
Example

Evaluating and Rewriting Logarithmic Expressions

Paulina has recently become excited learning about logarithms.

Paulina thinking of the definition of a logarithm

She eagerly went to her math teacher and asked for some introductory exercises to practice evaluating and rewriting logarithmic expressions. Help her get off to a good start!

a Evaluate the expression log_5 125.
b Rewrite the logarithmic equation log_4 a=x as an exponential equation.
c Rewrite the exponential equation 3^x=a as a logarithmic equation.

Hint
a To what power must the base of 5 be raised to obtain 125?
b Use the definition of a logarithm.
c How can an exponential equation be rewritten using a logarithm?

Solution
a To evaluate the given expression, it is helpful to recall what the definition of a logarithm is.

log_b a= c ⇔ b^c= a With this definition in mind, let x be the value of log_5 125. log_5 125= x ⇔ 5^x= 125 Given that 5 is the base, Paulina should ask herself what number it must be raised to in order to reach 125. Well, 5 to the power of 3 is equal to 125, the value of x is 3. Therefore, log_5 125=3. log_5 125= 3 ⇔ 5^3= 125

b Similar to Part A, to rewrite log_4 a=x as an exponential equation, the definition of a logarithm will be used.

log_b a= c ⇔ b^c= a Therefore, Paulina should substitute b= 4 and c= x in the above definition. log_4 a= x ⇔ 4^x= a

c To rewrite 3^x=a as a logarithmic equation, the definition of a logarithm will be used one last time.

b^c= a ⇔ log_b a= c Here, b= 3 and c= x will be substituted into the definition. 3^x= a ⇔ log_3 a= x

Pop Quiz

Evaluating Logarithms

Evaluate the logarithms.

evaluate the expression
Pop Quiz

Rewriting Logarithmic and Exponential Equations

Rewrite the logarithmic equations as exponential equations and the exponential equations as logarithmic equations.

rewrite the equation
Discussion

Properties of Logarithms

For the proof of these properties, two identities will be used. Start by recalling the definition of a logarithm. log_b a=c ⇔ a=b^c The first equation of the definition states that c=log_b a. Therefore log_b a can be substituted for c in the second equation. Furthermore, the second equation states that a is equal to b^c. This means that b^c can be substituted for a in the first equation.

log_b a=c ⇔ a=b^c
a=b^c Substitute a=b^(log_b a)
log_b a=c Substitute log_b b^c=c

With this information in mind, three properties can be stated.

Rule

Product Property of Logarithms

The logarithm of a product can be written as the sum of the individual logarithms of each factor.


log_b mn=log_b m+log_b n

This property is only valid for positive values of b, m, and n, and for b≠ 1. As an example, the expression log_3 (7*4) can be rewritten using this property. log_3 (7*4)=log_3 7+log_3 4

Proof

The obtained identities will be used together with the Product of Powers Property to prove the Product Property of Logarithms.
log_b mn
Rewrite

Rewrite mn as m* n

log_b(m* n)

m=b^(log_b(m))

log_b(b^(log_b m)* b^(log_b n))
MultPow

a^m*a^n=a^(m+n)

log_b(b^(log_b m+log_b n))

log_b(b^m)=m

log_b m+log_b n
Rule

Quotient Property of Logarithms

The logarithm of a quotient can be written as the difference between the logarithm of the numerator and the logarithm of the denominator.


log_b m/n=log_b m -log_b n

This property is valid for positive values of b, m, and n, and for b≠ 1. For example, the expression log_3 74 can be rewritten using this property. log_3 7/4=log_3 7-log_3 4

Proof

The obtained identities will be used together with the Quotient of Powers Property to prove the Quotient Property of Logarithms.
log_b m/n

m=b^(log_b(m))

log_b(b^(log_b m)/b^(log_b n))
DivPow

a^m/a^n= a^(m-n)

log_b(b^(log_b m-log_b n))

log_b(b^m)=m

log_b m-log_b n
Rule

Power Property of Logarithms

The logarithm of a power can be written as the product of the exponent and the logarithm of the base.


log_b m^n =nlog_b m

This property is valid for positive values of b, m, and n, and for b≠ 1. For example, log_2 7^4 can be rewritten using this property. log_2 7^4=4 log_2 7

Proof

The obtained identities will be used together with the Power of a Power Property to prove the Power Property of Logarithms.
log_b m^n

m=b^(log_b(m))

log_b (b^(log_b m) )^n
PowPow

(a^m)^n=a^(m* n)

log_b b^((log_b m )* n)

log_b(b^m)=m

(log_b m)* n
CommutativePropMult

Commutative Property of Multiplication

nlog_b m
Example

Using Properties to Evaluate Expressions

After understanding the definition of a logarithm and learning about its properties, Paulina is ready to delve deeper into this topic.
Paulina thinking about the properties of logarithms
As a part of her deep dive into properties of logarithms, she begins with two approximations. log_2 3≈ 1.585 and log_2 5≈ 2.322 Without using a calculator, only relying on the two given approximations, Paulina wants to evaluate three logarithmic expressions. This will make her feel like she is really understaning how to operate with logarithms. Help her find the answer! Write each answer rounded to three decimal places.
a log_2 15
b log_2 1/5
c log_2 24

Hint
a Use the Product Property of Logarithms.
b Use the Quotient Property of Logarithms.
c Use the Power Property of Logarithms.

Solution
a Consider the Product Property of Logarithms.
log_b mn = log_b m+log_b n Knowing that log_2 3≈ 1.585 and that log_2 5≈ 2.322, this property can be used to evaluate the given expression. Start by rewriting 15 as the product of 3 and 5.
log_2 6
SplitIntoFactors

Split into factors

log_2 (3* 5)

log_2(mn)=log_2(m) + log_2(n)

log_2 3+ log_2 5

log_2 3 ≈ 1.585, log_2 5 ≈ 2.322

1.585+ 2.322
AddTerms

Add terms

3.907
b Because of the quotient inside the logarithm, the Quotient Property of Logarithms will be used this time.
log_b m/n = log_b m-log_b n Also, as a consequence of the definition of a logarithm, it is known that log_b 1=0 for any positive number b different than 1. Using this information, the above property, and the fact that log_2 5≈ 2.322, the given expression can be evaluated.
log_2 1/5

log_2(a/b)=log_2(a) - log_2(b)

log_2 1-log_2 5

log_2(1) = 0

0-log_2 5

log_2 5 ≈ 2.322

0- 2.322
SubTerm

Subtract term

-2.322
c Recall the Power Property of Logarithms and the Product Property of Logarithms.
Power:& log_b m^n = nlog_b m Product:& log_b mn = log_b m + log_b n Also, as a consequence of the definition of a logarithm, it is known that log_b b=1 for any positive number b different than 1. Using this information, the above properties, and the fact that log_2 3≈ 1.585, the given expression can be evaluated.
log_2 24
SplitIntoFactors

Split into factors

log_2 (8* 3)

log_2(mn)=log_2(m) + log_2(n)

log_2 8 +log_2 3
WritePow

Write as a power

log_2 2^3 +log_2 3

log_2(a^m)= m* log_2(a)

3log_2 2 +log_2 3

log_2(2) = 1

3(1)+log_2 3
IdPropMult

Identity Property of Multiplication

3+log_2 3

log_2 3 ≈ 1.585

3+ 1.585
AddTerms

Add terms

4.585
Example

Using Properties to Expand Expressions

Paulina has moved beyond only understanding the definition of a logarithm. She can now rewrite exponential equations as logarithmic equations and convert logarithmic equations into exponential equations. On top of all of that, she can even evaluate logarithmic expressions. Paulina is starting to master this topic!
Paulina thinking about using the properties of logarithms to expand expressions that involve logarithms
Now, there are even more interesting challenges to face. This time, Paulina needs to expand two logarithmic expressions by using the Properties of Logarithms. Help her do this! Assume that all the variables involved are positive.
a log_5 25x^3/y^2
b log_4 sqrt(xy/4)

Hint
a Start by using the Quotient Property of Logarithms. Then, use the Product Property of Logarithms. Finally, use the Power Property of Logarithms.
b Start by rewriting the square root as a rational exponent. Then, use the Power Property of Logarithms to remove the exponent.

Solution
a Consider the Properties of Logarithms.
Product:& log_b mn=log_b m+log_b n Quotient:& log_b m/n=log_b m-log_b n Power:& log_b m^n=nlog_b m Here, b, m, and n are positive, where b≠ 1. The main operation in the given logarithmic expression is a division. Therefore, the first property that should be used is the Quotient Property of Logarithms. Then, the Product Property of Logarithms and the Power Property of Logarithms will be applied.
log_5 25x^3/y^2

log_5(a/b)=log_5(a) - log_5(b)

log_5 25x^3-log_5 y^2

log_5(mn)=log_5(m) + log_5(n)

log_5 25+log_5 x^3-log_5 y^2

log_5(a^m)= m* log_5(a)

log_5 25+3log_5 x-2log_5 y

Calculate logarithm

2+3log_5 x-2log_5 y
b To use the Power Property of Logarithms, the square root will be written as a rational exponent.
log_4 sqrt(xy/4) = log_4 (xy/4)^(12) Now, the mentioned property can be used. Then, the Quotient Property of Logarithms and the Product Property of Logarithms can also be used.
log_4 (xy/4)^(12)

log_4(a^m)= m* log_4(a)

1/2log_4 xy/4

log_4(a/b)=log_4(a) - log_4(b)

1/2(log_4 xy-log_4 4)

log_4(mn)=log_4(m) + log_4(n)

1/2(log_4 x+log_4 y-log_4 4)
Distr

Distribute 1/2

1/2 log_4 x+1/2 log_4 y-1/2log_4 4

log_4(4) = 1

1/2 log_4 x+1/2 log_4 y-1/2(1)
IdPropMult

Identity Property of Multiplication

1/2 log_4 x+1/2 log_4 y-1/2
Example

Using Properties to Condense Expressions

Paulina feels extremely confident about her skills in using logarithms! Now, instead of expanding algebraic and numeric expressions that involve logarithms, she will practice condensing them.
Paulina thinking about using the properties of logarithms to condense expressions that involve logarithms
To finally master logarithmic expressions with and without variables, Paulina wants to condense two expressions using the Properties of Logarithms. Help her do this! Assume that all variables involved are positive.
a log_2 12 +3log_2 5-log_2 6
b 5log_3 2-6log_3 x+2log_3 y

Hint
a Use the Product Property of Logarithms, the Power Property of Logarithms, and the Quotient Property of Logarithms.
b Use the Properties of Logarithms.

Solution
a To condense this expression, three properties of logarithms will be recalled first. These are the Product Property of Logarithms, the Quotient Property of Logarithms, and the Power Property of Logarithms.
Product:& log_b mn=log_b m+log_b n Quotient:& log_b m/n=log_b m-log_b n Power:& log_b m^n=nlog_b m These three properties will be used to condense the expression.
log_2 12 +3log_2 5-log_2 6

m* log_2(a)=log_2(a^m)

log_2 12 +log_2 5^3-log_2 6
CalcPow

Calculate power

log_2 12 +log_2 125-log_2 6

log_2(m) + log_2(n)=log_2(mn)

log_2 (12 * 125)-log_2 6
Multiply

Multiply

log_2 1500-log_2 6

log_2(m) - log_2(n)=log_2(m/n)

log_2 1500/6
CalcQuot

Calculate quotient

log_2 250
b Similar to Part A, to condense this expression, the same three Properties of Logarithms will be applied.
5log_3 2-6log_3 x+2log_3 y

m* log_3(a)=log_3(a^m)

log_3 2^5-log_3 x^6+log_3 y^2
CalcPow

Calculate power

log_3 32-log_3 x^6+log_3 y^2

log_3(m) - log_3(n)=log_3(m/n)

log_3 32/x^6+log_3 y^2

log_3(m) + log_3(n)=log_3(mn)

log_3 (32/x^6* y^2)
MoveRightFacToNum

a/c* b = a* b/c

log_3 32y^2/x^6
Closure

Inverse Operations

In this lesson, it has been presented that exponents and logarithms are their own inverses. This means that these two operations essentially undo each other.
inverse of logarithm is raising to the power of x, and inverse of raising to the power of x is a logarithm
The definition of a logarithm and important Properties of Logarithms have also been taught and practiced in this lesson.
Definition log_b a=c ⇔ a=b^c
Identity Derived From the Definition log_b b=1
Identity Derived From the Definition log_b 1=0
Product Property of Logarithms log_b mn = log_b m+log_b n
Quotient Property of Logarithms log_b m/n = log_b m-log_b n
Power Property of Logarithms log_b m^n = nlog_b m

It is important to keep in mind that these properties are only valid for positive values of a, b, m, and n, where b≠ 1. Furthermore, these properties can be used in several situations.

  • Writing a power as a logarithm and writing a logarithm as a power.
  • Evaluating logarithms.
  • Using given approximations of certain logarithms to approximate other logarithmic expressions.
  • Expanding and condensing logarithmic expressions with and without variables.


Level 1
Level 2
Level 3
Properties of Logarithms
Exercise 3.1
Simplify the following logarithmic expression. log_(sqrt(a)) a^2, a>0, a≠ 1 Write the answer in exact form.

Let's start by using the definition of a logarithm to rewrite the given logarithmic expression as an exponential expression. To do this, let c be the numerical value of log_(sqrt(a)) a^2. Definition:& log_b m = n ⇔ b^n= m Expression:& log_(sqrt(a)) a^2 = c ⇔ ( sqrt(a))^c= a^2 We found that sqrt(a) raised to the power of c equals a^2. To solve the obtained equation for c, we will rewrite sqrt(a) as a^(12) and use the Power of a Power Property. Let's do it!

Finally, we will use the Property of Equality for Exponential Equations to equate the exponents.

E
Hint & Answer
S
Solution
Simplify the logarithmic expression. x^(3log_x 2 - log_x 5) Write the exact numerical value.

Let's start simplifying the given expression by using the Power Property of Logarithms.

Next, we can use the Quotient Property of Logarithms to simplify the exponent a bit more.

Finally, we can use the Inverse Property of Logarithms to eliminate the base of the power and the logarithm. x^(log_x 85)=8/5

E
Hint & Answer
S
Solution

Properties of Logarithms

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