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Logarithms of products

For positive factors in one valid logarithm base, the logarithm of a product equals the sum of the factors’ logarithms.

1 min read1 figureSources & further reading

On this page 6 sections

Overview
A logarithm turns multiplication of positive factors into addition
The rule can also collect a sum of logs into one log of a product
Key takeaway
Sources & further reading
Concept connections

Figure 1Base-two logarithms: log₂4=2 and log₂8=3 concatenate to log₂32=5. The adjacent spans have lengths in the ratio 2:3.

01A logarithm turns multiplication of positive factors into addition#

A logarithm turns multiplication of positive factors into addition. In base $2$ , $4=2^2$ and $8=2^3$ .

Multiplying adds their exponents.

4\cdot8=2^2\cdot2^3=2^5

The logarithms are $\log_2 4=2$ and $\log_2 8=3$ . Their sum is the logarithm of the product $4\cdot8=32$ .

The product’s logarithm is the sum.

\log_2(32)=2+3=5

The same rule works in any one base $b>0$ with $b\ne1$ . Both factors must be positive so that both real logarithms exist.

The product rule.

\begin{gathered}\log_b(uv)\\=\log_bu+\log_bv\end{gathered}

Check your reasoning

Given $\log_3u=2$ and $\log_3v=4$ for positive $u,v$ , what is $\log_3(uv)$ ?

Show answer and explanation

The common base and positive factors allow addition: $2+4=6$ .

02The rule can also collect a sum of logs into one log of a product#

The rule can also collect a sum of logs into one log of a product. It does not split a sum inside the argument.

Each $\log_2 1$ is $0$ , so their sum is $0$ . But:

\log_2(1+1)=1

A positive product alone is not enough to split it. The product $(-2)(-4)=8$ has a real logarithm, but the individual real logarithms of $-2$ and $-4$ do not exist. The rule needs positive factors individually.

Check your reasoning

For $u,v>0$ , combine $\log_{10}u+\log_{10}v$ .

A $\log_{10}(u+v)$
B $\log_{10}(uv)$
C $\log_{10}u\cdot\log_{10}v$

Show answer and explanation

$\log_{10}(uv)$

Add logs by multiplying their arguments.

Key takeaway

A product inside a logarithm becomes addition outside. Use the same base and positive factors; the rule does not split a sum inside the argument.

Rewrite a positive product using logarithms

Sources & further reading

[1]
OpenStax College Algebra 2e, 6.5 ↗OpenStax · Book

01A logarithm turns multiplication of positive factors into addition#

02The rule can also collect a sum of logs into one log of a product#

A product inside a logarithm becomes addition outside. Use the same base and positive factors; the rule does not split a sum inside the argument.

Sources & further reading

Reference this concept