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Applying Properties of Logarithms to Rewrite Logarithmic Expressions

Author: Sophia
PDF Version

what's covered
In this lesson, you will apply properties in order to expand and condense logarithmic functions. Specifically, this lesson will cover:

Table of Contents

1. Properties of Logarithms

1a. Inverse Properties

Since f open parentheses x close parentheses equals b to the power of x and g open parentheses x close parentheses equals log subscript b x are inverse functions, it follows that open parentheses f ring operator g close parentheses open parentheses x close parentheses equals x and open parentheses g ring operator f close parentheses open parentheses x close parentheses equals x.

Computing open parentheses f ring operator g close parentheses open parentheses x close parentheses colon

open parentheses f ring operator g close parentheses open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses This is the definition of composition of f with g.
equals f open parentheses log subscript b x close parentheses Substitute g open parentheses x close parentheses equals log subscript b x.
equals b to the power of log subscript b x end exponent Substitute log subscript b x for x in the function f.

Since open parentheses f ring operator g close parentheses open parentheses x close parentheses equals x comma it follows that b to the power of log subscript b x end exponent equals x.

Computing open parentheses g ring operator f close parentheses open parentheses x close parentheses colon

open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses This is the definition of composition of g with f.
equals g open parentheses b to the power of x close parentheses Substitute f open parentheses x close parentheses equals b to the power of x.
equals log subscript b open parentheses b to the power of x close parentheses Substitute b to the power of x for x in the function g.

Since open parentheses g ring operator f close parentheses open parentheses x close parentheses equals x comma it follows that log subscript b open parentheses b to the power of x close parentheses equals x. We actually defined this property earlier.

formula to know
Inverse Properties of Logarithms
Given base b, where b greater than 0 and b not equal to 1:
log subscript b open parentheses b to the power of x close parentheses equals x
b to the power of log subscript b x end exponent equals x

EXAMPLE

Evaluate the following expressions.

Expression Evaluation
10 to the power of log 5 end exponent Since log     5 uses base 10, we can use the property b to the power of log subscript b x end exponent equals x, where b equals 10.

Then, 10 to the power of log 5 end exponent equals 5.
log subscript 6 open parentheses 6 to the power of 3 x end exponent close parentheses We can use the property log subscript b open parentheses b to the power of x close parentheses equals x, with b equals 6.

Then, log subscript 6 open parentheses 6 to the power of 3 x end exponent close parentheses equals 3 x.
e to the power of ln square root of 2 end exponent Since ln   square root of 2 uses base e, we can use the property b to the power of log subscript b x end exponent equals x, where b equals e.

Then, e to the power of ln square root of 2 end exponent equals square root of 2.

try it
Consider the expression e to the power of ln 0.65 end exponent.
Use properties to simplify the expression.
0.65

try it
Consider the expression log   open parentheses 10 to the power of short dash b squared end exponent close parentheses.
Use properties to simplify the expression.
short dash b squared

try it
Consider the expression e to the power of ln open parentheses 25 t close parentheses end exponent.
Use properties to simplify the expression.
25 t

1b. Logarithms of Products, Quotients, and Powers

Recall the following properties of exponents.

Function Properties
b to the power of x times b to the power of y equals b to the power of x plus y end exponent When multiplying two numbers with the same base, add the exponents.
b to the power of x over b to the power of y equals b to the power of x minus y end exponent When dividing two numbers with the same base, subtract the exponents.
open parentheses b to the power of x close parentheses to the power of y equals b to the power of x y end exponent When raising b to the power of x to a power, multiply the exponents.

To help establish logarithm properties, consider the following.

Suppose log subscript b x equals M and log subscript b y equals N comma where x greater than 0 comma y greater than 0, and M and N are real numbers. Rewriting each in exponential form, b to the power of M equals x and b to the power of N equals y.

Now, consider the expression log subscript b open parentheses x y close parentheses.

log subscript b open parentheses x y close parentheses equals log subscript b open parentheses b to the power of M b to the power of N close parentheses Substitute x equals b to the power of M and y equals b to the power of N.
equals log subscript b open parentheses b to the power of M plus N end exponent close parentheses Apply the property b to the power of M b to the power of N equals b to the power of M plus N end exponent.
equals M plus N Apply the inverse property, log subscript b open parentheses b to the power of n close parentheses equals n.
equals log subscript b x plus log subscript b y Substitute M equals log subscript b x and N equals log subscript b y.

This gives us the property log subscript b open parentheses x y close parentheses equals log subscript b x plus log subscript b y.

In other words, the logarithm of a product is equal to the sum of the logarithms of its individual factors, provided that each factor is positive.

There are similar properties for log subscript b open parentheses x over y close parentheses and log subscript b open parentheses x to the power of y close parentheses comma which are given below without proof.

formula to know
Product Property
log subscript b open parentheses x y close parentheses equals log subscript b x plus log subscript b y
Quotient Property
log subscript b open parentheses x over y close parentheses equals log subscript b x minus log subscript b y
Power Property
log subscript b open parentheses x to the power of y close parentheses equals y times log subscript b x

These properties are used to rewrite logarithmic expressions in two ways:

  • Expand a single logarithm as a sum, difference, or multiple of logarithms.
  • Write an expanded logarithmic expression as a single logarithm.
We will explore this in the next two sections.

2. Expanding Logarithmic Expressions

Expanding a logarithmic expression means to write it as a sum, difference, or multiple of logarithmic expressions. These next examples will help illustrate what to look for when the expanded form of logarithms is desired.

step by step
  1. Apply the product and quotient properties first to “break up” the expression into a sum/difference.
  2. Apply the power property where relevant.
  3. Perform any relevant simplifications.

EXAMPLE

Use logarithm properties to expand the expression ln   open parentheses fraction numerator 2 x over denominator y end fraction close parentheses.

ln   open parentheses fraction numerator 2 x over denominator y end fraction close parentheses This is the expression.
equals ln   open parentheses 2 x close parentheses minus ln     y fraction numerator 2 x over denominator y end fraction is a quotient; apply the quotient property.
equals ln     2 plus ln     x minus ln     y 2 x is a product; apply the product property.

Thus, in expanded form, ln   open parentheses fraction numerator 2 x over denominator y end fraction close parentheses equals ln     2 plus ln     x minus ln     y.

Here is another example in which the power property is used.

EXAMPLE

Use logarithm properties to expand the expression log   open parentheses x squared y to the power of 4 close parentheses.

log   open parentheses x squared y to the power of 4 close parentheses This is the original expression.
equals log   open parentheses x squared close parentheses plus log   open parentheses y to the power of 4 close parentheses x squared y to the power of 4 is a product; apply the product property.
equals 2   log     x plus 4   log     y Apply the power property.

Thus, in expanded form, log   open parentheses x squared y to the power of 4 close parentheses equals 2   log     x plus 4   log     y.

try it
Consider the expression log subscript 4 open parentheses fraction numerator 2 x over denominator y cubed end fraction close parentheses.
Use logarithm properties to expand this expression.
log subscript 4 open parentheses fraction numerator 2 x over denominator y cubed end fraction close parentheses Original expression.
equals log subscript 4 open parentheses 2 x close parentheses minus log subscript 4 open parentheses y cubed close parentheses Use the quotient property.
equals log subscript 4 2 plus log subscript 4 x minus log subscript 4 open parentheses y cubed close parentheses 2x is a product, so apply the product property.
equals log subscript 4 2 plus log subscript 4 x minus 3   log subscript 4 y Apply the power property to write log subscript 4 y cubed equals 3 log subscript 4 y.

Thus, log subscript 4 open parentheses fraction numerator 2 x over denominator y cubed end fraction close parentheses equals log subscript 4 2 plus log subscript 4 x minus 3   log subscript 4 y.

watch
In this video, you’ll use properties of logarithms to write ln   square root of fraction numerator x cubed over denominator y minus 6 end fraction end root in expanded form.

3. Condensing Logarithmic Expressions

To condense a logarithmic expression into a single logarithm, apply the properties as we did when expanding expressions, but in reverse.

step by step
  1. Reverse the power property first for any expressions: y times log subscript b x equals log subscript b open parentheses x to the power of y close parentheses
  2. Reverse the sum/difference properties: log subscript b x plus log subscript b y equals log subscript b open parentheses x y close parentheses or log subscript b x minus log subscript b y equals log subscript b open parentheses x over y close parentheses

EXAMPLE

Use logarithm properties to write 3   log subscript 4 x plus log subscript 4 5 minus 2   log subscript 4 z as a single logarithm.

3   log subscript 4 x plus log subscript 4 5 minus 2   log subscript 4 z This is the original expression.
equals log subscript 4 x cubed plus log subscript 4 5 minus log subscript 4 z squared Reverse the power property.
equals log subscript 4 open parentheses 5 x cubed close parentheses minus log subscript 4 z squared Reverse the product property.
equals log subscript 4 open parentheses fraction numerator 5 x cubed over denominator z squared end fraction close parentheses Reverse the quotient property.

As a single logarithm, 3   log subscript 4 x plus log subscript 4 5 minus 2   log subscript 4 z equals log subscript 4 open parentheses fraction numerator 5 x cubed over denominator z squared end fraction close parentheses.

try it
Consider the expression 2   ln     x minus 3   ln     y plus 4   ln   open parentheses z plus 1 close parentheses.
Write this expression as a single logarithm.
2   ln     x minus 3   ln     y plus 4   ln   open parentheses z plus 1 close parentheses This is the original expression.
equals ln     x squared minus ln     y cubed plus ln   open parentheses z plus 1 close parentheses to the power of 4 Reverse the power property.
equals ln   open parentheses x squared over y cubed close parentheses plus ln   open parentheses z plus 1 close parentheses to the power of 4 Reverse the quotient property.
equals ln   open square brackets fraction numerator x squared open parentheses z plus 1 close parentheses to the power of 4 over denominator y cubed end fraction close square brackets Reverse the product property. Recall that when multiplying a fraction to an expression, the expression ends up in the numerator.

Thus, as one single logarithm, 2   ln     x minus 3   ln     y plus 4   ln   open parentheses z plus 1 close parentheses equals ln   open square brackets fraction numerator x squared open parentheses z plus 1 close parentheses to the power of 4 over denominator y cubed end fraction close square brackets.

4. The Change of Base Formula

Consider the logarithm log subscript 2 40. Since the base is not e or 10, and 40 is not a recognizable power of 2, we have no way to evaluate this logarithm without using graphs.

Let y equals log subscript 2 40. Then, this equation in exponential form is 2 to the power of y equals 40.

2 to the power of y equals 40 This is the original equation.
ln   open parentheses 2 to the power of y close parentheses equals ln     40 Apply the natural logarithm to both sides. This is allowed through the one-to-one property.
y open parentheses ln     2 close parentheses equals ln     40 Use the power property of logarithms.
y equals fraction numerator ln   40 over denominator ln   2 end fraction Divide both sides by ln     2.

This means that we have two different expressions for y; therefore, they must be equal. That is, y equals log subscript 2 40 equals fraction numerator ln   40 over denominator ln   2 end fraction.

Instead of the natural logarithm, we could have applied the common logarithm to both sides. The result would be similar: log subscript 2 40 equals fraction numerator log   40 over denominator log   2 end fraction

In fact, if you evaluate both fraction numerator ln   40 over denominator ln   2 end fraction and fraction numerator log   40 over denominator log   2 end fraction using your calculator, you get the same result, which is 5.321928094887362…

Since 2 to the power of 5 equals 32 and 2 to the power of 6 equals 64 comma this answer makes sense. Also, 2 to the power of 5.321928094887362 end exponent almost equal to 40 comma which checks.

While not usually convenient, a logarithm in any other base could be applied to both sides. Base 10 and e are used since these are the only bases that allow us to approximate logarithms with direct keys on most calculators.

formula to know
Change of Base Formula
log subscript b x equals fraction numerator ln     x over denominator ln     b end fraction equals fraction numerator log     x over denominator log     b end fraction equals fraction numerator log subscript c x over denominator log subscript c b end fraction comma where x greater than 0 comma b greater than 0, and b not equal to 1 comma and c greater than 0 and c not equal to 1.

try it
Consider the logarithm log subscript 5 189.
Approximate the value of the logarithm to the nearest thousandth.
log subscript 5 189 equals fraction numerator log   189 over denominator log   5 end fraction almost equal to 3.257

Note: If using natural logarithms, we have fraction numerator ln   189 over denominator ln   5 end fraction almost equal to 3.257.

summary
In this lesson, you learned about the properties of logarithms, understanding that the inverse properties of logarithms are used to simplify expressions, while properties for logarithms of products, quotients, and powers are used to expand and condense logarithmic expressions. You learned that expanding a logarithmic expression means to write it as a sum, difference, or multiple of logarithmic expressions, while condensing a logarithmic expression into a single logarithm involves applying the properties as we did when expanding expressions, but in reverse. Finally, you learned that the change of base formula is used to evaluate logarithms that are neither common nor natural logarithms.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Formulas to Know
Change of Base Formula

log subscript b x equals fraction numerator ln     x over denominator ln     b end fraction equals fraction numerator log     x over denominator log     b end fraction equals fraction numerator log subscript c x over denominator log subscript c b end fraction comma where x greater than 0 comma b greater than 0 comma and b not equal to 1 comma and c greater than 0 and c not equal to 1.

Inverse Properties of Logarithms

Given base b, where b greater than 0 and b not equal to 1 colon
log subscript b open parentheses b to the power of x close parentheses equals x
b to the power of log subscript b x end exponent equals x

Power Property

log subscript b open parentheses x to the power of y close parentheses equals y times log subscript b x

Product Property

log subscript b open parentheses x y close parentheses equals log subscript b x plus log subscript b y

Quotient Property

log subscript b open parentheses x over y close parentheses equals log subscript b x minus log subscript b y

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