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All courses Algebra II · F-LE.4.3 39 of 55
Use logarithm properties to simplify numeric logarithmic expressions and estimate values.

Combine a sum of logarithms using the product law

Problem
What is a simplified form of \(\log_2(4)+\log_2(8)\) using the product law of logarithms?
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Hint

Check that both logarithms have the same base before using the product law.

The product law keeps base 2 and multiplies the arguments. After combining, check whether the product is a power of 2.

Solution walkthrough

01

Apply the product law

\[\log_2(4)+\log_2(8)=\log_2(4*8)=\log_2(32)\]

A sum of same-base logarithms becomes the logarithm of the product of their positive arguments.

02

Recognize the exact power

\[32=2^5\]

The condensed argument is an exact fifth power of the logarithm base.

03

State the simplified form and value

\[\log_2(32)=5\]

Log base 2 of 32 asks for the exponent 5, so this gives both the requested product-law form and its exact evaluation.

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Another way

  1. Evaluate separately: log base 2 of 4 is 2 and log base 2 of 8 is 3, so the sum is 5.

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Common mistake

Do not add the arguments inside the logarithm. A sum of logarithms uses the product 4 times 8, not 4 plus 8.

Solution walkthrough video