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All courses Math III · F-LE.4.3 32 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 log₂(4)+log₂(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

Check the bases

\[\log_2(4)+\log_2(8):~\text{both}~\text{bases}~\text{are}~2\]

The product law applies because both logarithms use base 2. Matching bases are required before the logs can be combined this way.

02

Apply the product law

\[\log_b(M)+\log_b(N)=\log_b(MN)\]

For a sum of logs with the same base, keep the base and multiply the arguments inside one logarithm.

03

Multiply the arguments

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

Here the two arguments are 4 and 8, so the inside product is 32.

04

Evaluate the combined log

\[2^5=32,~\text{so}~\log_2(32)=5\]

A logarithm asks for the exponent on the base. Since 2 raised to the fifth power is 32, the logarithm equals 5.

05

State the answer

\[\text{Simplified}~\text{form}:~\log_2(32)=5\]

This matches the question because it uses the product law to combine the logs and then evaluates the exact power of 2.

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Another way

  1. Evaluate first: log₂(4)=2 and log₂(8)=3, so the original sum is 2+3=5.

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Common mistake

A common mistake is writing log₂(4+8). The product law says to multiply the arguments, so 4 and 8 become 4*8, not 4+8.