Properties of Logarithmic Functions

 

Learning Objective(s)

·         Express the logarithm of a product as a sum of logarithms.

·         Express the logarithm of a quotient as a difference.

·         Express the logarithm of a power as a product.

·         Simplify logarithmic expressions.

 

Introduction

 

Throughout your study of algebra, you have come across many properties—such as the commutative, associative, and distributive properties. These properties help you take a complicated expression or equation and simplify it.

 

The same is true with logarithms. There are a number of properties that will help you simplify complex logarithmic expressions. Since logarithms are so closely related to exponential expressions, it is not surprising that the properties of logarithms are very similar to the properties of exponents. As a quick refresher, here are the exponent properties.

 

Properties of Exponents

 

Product of powers:

Quotient of powers:

Power of a power:

 

 

One important but basic property of logarithms is logb bx = x. This makes sense when you convert the statement to the equivalent exponential equation. The result? bx = bx.

 

Let’s find the value of y in. Remember , so means and y must be 2, which means . You will get the same answer that  equals 2 by using the property that logb bx = x.

 

Logarithm of a Product

 

Remember that the properties of exponents and logarithms are very similar. With exponents, to multiply two numbers with the same base, you add the exponents. With logarithms, the logarithm of a product is the sum of the logarithms.

 

 

Logarithm of a Product

 

The logarithm of a product is the sum of the logarithms:

logb (MN) = logb M + logb N

 

 

 

Let’s try the following example.

 

 

Example

Problem

Use the product property to rewrite .

 

 

 

 

Use the product property to write as a sum.

 

Simplify each addend, if possible. In this case, you can simplify both addends.

Rewrite log2 4 as log2 22and log2 8 as log2 23, then use the property logb bx = x.

Or, rewrite log2 4 = y as 2y = 4 to find y = 2, and log2 8 = y as 2y = 8 to find y = 3.

Use whatever method makes sense to you.

Answer

 

 

 

Another way to simplify would be to multiply 4 and 8 as a first step.

 

 

You get the same answer as in the example!

 

Notice the similarity to the exponent property: bmbn = bm + n, while logb (MN) = logb M + logb N. In both cases, a product becomes a sum.

 

 

Example

Problem

Use the product property to rewrite log3 (9x).

 

log3 (9x) = log3 9 + log3 x

Use the product property to write as a sum.

 

log3 9 + log3 x =

log3 32 + log3 x =

2 + log3 x

Simplify each addend, if possible. In this case, you can simplify log3 9 but not log3 x.

Rewrite log3 9 as log3 32, then use the property logb bx = x.

Or, simplify log3 9 by converting log3 9 = y to 3y = 9 and finding that y = 2.

 

Use whatever method makes sense to you.

Answer

log3(9x) = 2 + log3 x

 

 

 

If the product has many factors, you just add the individual logarithms:

logb (ABCD) = logb A + logb B + logb C + logb D.

 

 

Rewrite log2 8a, then simplify.

 

A) 3 log2 a

B) log2 3a

C) log2 (3 + a)

D) 3 + log2 a

 

Show/Hide Answer

A) 3 log2 a

Incorrect. The individual logarithms must be added, not multiplied. The correct answer is 3 + log2 a.

 

B) log2 3a

Incorrect. You found that log2 8 = 3, but you must first apply the logarithm of a product property. The correct answer is 3 + log2 a.

 

C) log2 (3 + a)

Incorrect. The logarithm of a product property says you separate the 8 and a into separate logarithms. The correct answer is 3 + log2 a.

 

D) 3 + log2 a

Correct. The logarithm of a product property says log2 8a = log2 8 + log2 a, and log2 8 = 3.

 

 

 

Logarithm of a Quotient

 

You can use the similarity between the properties of exponents and logarithms to find the property for the logarithm of a quotient. With exponents, to multiply two numbers with the same base, you add the exponents. To divide two numbers with the same base, you subtract the exponents. What do you think the property for the logarithm of a quotient will look like?

 

As you may have suspected, the logarithm of a quotient is the difference of the logarithms.

 

Logarithm of a Quotient

 

 

With both properties:  and, a quotient becomes a difference.

 

 

Example

 

Problem

Use the quotient property to rewrite .

 

 

log2  = log2 x – log2 2

Use the quotient property to rewrite as a difference.

Answer

= log2 x – 1

The first expression can’t be simplified further. However, the second expression can be simplified. What exponent on the base (2) gives a result of 2? Since 21 = 2, you know log2 2 = 1.

 

 

Which of these is equivalent to: .

 

A) 4 – log3 a

B)

C) log3 (4 – a)

D)

Show/Hide Answer

A) 4 – log3 a

Correct. The logarithm of a quotient property states , and log3 81 = 4.

 

B)

Incorrect. The individual logarithms must be subtracted, not divided. The correct answer is 4 – log3 a.

 

C) log3 (4 – a)

Incorrect. The logarithm of a quotient property says you separate the 81 and a into separate logarithms. The correct answer is 4 – log3 a.

 

D)

Incorrect. You found that log3 81 = 4, but you must first apply the logarithm of the quotient property. The correct answer is 4 – log3 a.

 

 

Logarithm of a Power

The remaining exponent property was power of a power: . The similarity with the logarithm of a power is a little harder to see.

 

Logarithm of a Power

 

 

With both properties,  and, the power “n” becomes a factor.

 

 

Example

Problem

Use the power property to simplify log3 94.

 

log3 94 = 4 log3 9

 

You could find 94, but that wouldn’t make it easier to simplify the logarithm. Use the power property to rewrite log3 94 as 4log3 9.

 

4 log3 9 = 4•2

You may be able to recognize by now that since 32 = 9, log3 9 = 2.

Answer

log3 94 = 8

Multiply the factors.

 

 

Notice in this case that you also could have simplified it by rewriting it as 3 to a power: log3 94 = log3 (32)4. Using exponent properties, this is log3 38 and by the property logb bx = x, this must be 8!

 

 

Example

Problem

Use the properties of logarithms to rewrite log4 64x.

 

 

 

 

 

Use the power property to rewrite  as .

 

Rewrite  as, then use the property  to simplify .

Or, you may be able to recognize by now that since 43 = 64, log464 = 3.

 

 

 

Answer

log4 64x = 3x

Multiply the factors.

 

 

Which of these is equivalent to:  log2 x8.

 

A) log2 3x

B) 8 log2 x

C) log2 8x

D) 3 log2 x

 

Show/Hide Answer

A) log2 3x

Incorrect. The exponent becomes a factor outside the logarithm. The correct answer is

8 log2 x.

 

B) 8 log2 x

Correct. By the power property, log2 x8 = 8log2 x. You can’t simplify this further.

 

C) log2 8x

Incorrect. The exponent becomes a factor outside the logarithm. The correct answer is

8 log2 x.

 

D) 3 log2 x

Incorrect. You probably noticed that log2 8 = 3, so you used 3 instead of 8 when you pulled the exponent out to be a factor. However, the exponent must be pulled outside the logarithm to be a factor without any other changes. The correct answer is 8 log2 x.

 

 

 

Simplifying Logarithmic Expressions

 

The properties can be combined to simplify more complicated expressions involving logarithms.

 

 

Example

Problem

Use the properties of logarithms to expand  into four simpler terms.

 

 

 

 

Use the quotient property to rewrite  as a difference of logarithms.

 

Now you have two logarithms, each with a product. Apply the product rule to each.

 

Be careful with the subtraction! Since all of log10 cd is subtracted, you have to subtract both parts of the term, (log10 c + log10 d).

Answer

 = log10 a + log10 b – log10 c – log10 d

 

 

 

Example

Problem

Simplify log6 (ab)4, writing it as two separate terms.

 

 

 

 

Use the power property to rewrite log6 (ab)4 as

4 log6 (ab).

 

You are taking the log of a product, so apply the product property.

 

Be careful: the value 4 is multiplied by the whole logarithm, so use parentheses when you rewrite log6 (ab) as (log6 a + log6 b)

Answer

log6 (ab)4  = 4 log6 a + 4 log6 b

 

Use the distributive property.

 

 

Simplify log3 x2y.

 

A) 2(log3 x + log3 y)

B) log3 x2 + log3 y

C) 2 log3 xy

D) 2 log3 x + log3 y

 

Show/Hide Answer

A) 2(log3 x + log3 y)

Incorrect. You may have started incorrectly by applying the power property, or you may have started correctly with the product property but then incorrectly applied the power property. The correct answer is 2 log3 x + log3 y.

 

B) log3 x2 + log3 y

Incorrect. While you correctly applied the product property first, log3x2 can be simplified further. The correct answer is 2 log3 x + log3 y.

 

C) 2 log3 xy

Incorrect. You probably started incorrectly by applying the power property. Start with the product property. The correct answer is 2 log3 x + log3 y.

 

D) 2 log3 x + log3 y

Correct. log3 x2y = log3 x2 + log3 y = 2 log3 x + log3 y.

 

 

Summary

Like exponents, logarithms have properties that allow you to simplify logarithms when their inputs are a product, a quotient, or a value taken to a power. The properties of exponents and the properties of logarithms have similar forms.

 

 

Exponents

Logarithms

Product Property

Quotient Property

Power Property

 

Notice how the product property leads to addition, the quotient property leads to subtraction, and the power property leads to multiplication for both exponents and logarithms.