Product Rule of Logarithms
Simple Explanation
The logarithm of a product equals the sum of the logarithms: log_b(MN) = log_b M + log_b N. This mirrors the exponent product rule, since logarithms are exponents in disguise.
Why Do We Need It?
The product rule turns multiplication inside a logarithm into addition outside it — the historical reason logarithms were invented, to make hand multiplication of large numbers as easy as addition.
Formula
Product Rule of Logarithms
log_b(MN) = log_b M + log_b N
The logarithm of a product equals the sum of the logarithms of its factors.
- b
- — the base of the logarithm (b > 0, b ≠ 1)
- M, N
- — positive real numbers being multiplied
When to use it: Whenever you need to expand the logarithm of a product into a sum, or condense a sum of logarithms into one logarithm of a product.
Worked Example
Expand a logarithm of a product
Expand log₂(8 × 16) using the product rule, then evaluate.
Why Does This Work?
If log_b M = p and log_b N = q, then M = bᵖ and N = bᵠ, so MN = bᵖ·bᵠ = bᵖ⁺ᵠ by the exponent product rule — meaning log_b(MN) is exactly p + q, i.e. log_b M + log_b N.
Real-Life Example
Combining sound intensities in decibels
Two sound sources combine their intensities, and decibel levels are logarithmic.
Because decibels are proportional to a logarithm, combining sound sources involves adding logarithmic (decibel-like) quantities rather than the raw intensities directly.
Practice
Which expression equals log₃(9x)?
MediumCommon mistake
Turning log_b(MN) into (log_b M)(log_b N), i.e. multiplying the logs instead of adding them — the product rule converts multiplication to addition, not to another multiplication.
Quick Review
- log_b(MN) = log_b M + log_b N.
- The product rule lets you expand a log of a product into a sum, or condense a sum into a log of a product.
- It comes directly from the exponent product rule, since logs are exponents.