Logarithm Quotient Rule
The log of a quotient is the difference of the logs.
Formula
Variables
Example
log(10/1)=1
How it works
A logarithm tells you the exponent needed to produce a number, so dividing two numbers with the same base can be rewritten as subtracting their exponents. Since x/y = x \u00d7 y^{-1}, taking a log turns the product into a sum, and the reciprocal into a negative sign, which is why the quotient becomes a difference. This rule only works when x and y are positive and the base b is valid for logarithms (b > 0 and b \u2260 1).
When to use it
Use this when simplifying logarithmic expressions that contain division, especially in algebra problems where you want to expand or combine logs. It is also useful in science and engineering when working with ratios, such as comparing signal strengths, concentrations, or scales expressed with logarithms.
Worked example, step by step
Evaluate \log_2(32/4) \log_2(32/4) = \log_2 32 - \log_2 4 \log_2 32 = 5 because 2^5 = 32 \log_2 4 = 2 because 2^2 = 4 So, \log_2(32/4) = 5 - 2 \log_2(32/4) = 3
Common mistakes to avoid
- Subtracting the arguments instead of the logarithms, such as writing log_b(x/y) = log_b x - y
- Applying the rule when x or y is zero or negative, which is not allowed for real logarithms
- Forgetting that the base must be the same in every logarithm before using the quotient rule
Did You Know?
Logarithms were invented by John Napier in 1614 specifically to turn tedious division into simple subtraction.
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