Logarithm Product Rule
The log of a product is the sum of the logs.
Formula
Variables
Example
log(100) = log(10)+log(10) = 2
How it works
Logs turn multiplication into addition because they answer the question “what exponent of b gives this number?” If x = b^m and y = b^n, then xy = b^(m+n), so taking the log base b of both sides gives log_b(xy) = m+n = log_b x + log_b y. This rule works only when the log base is the same on both terms and the arguments are positive.
When to use it
Use this rule when a logarithm contains a product that is hard to simplify directly, especially in algebra, calculus, and solving exponential equations. It is also useful when expanding or condensing log expressions.
Worked example, step by step
Use the product rule to simplify log_2(8·4) log_2(8·4) = log_2(8) + log_2(4) log_2(8) = 3 because 2^3 = 8 log_2(4) = 2 because 2^2 = 4 So, log_2(8·4) = 3 + 2 = 5
Common mistakes to avoid
- Adding the numbers inside the logarithm instead of splitting the product into a sum of logs
- Using the product rule when the logs have different bases
- Forgetting that x and y must be positive for the logarithms to be defined
Did You Know?
Turning multiplication into addition, this log law powered slide rules that guided engineers to the Moon.
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