The three logarithm laws convert products, quotients, and powers inside a logarithm into addition, subtraction, and multiplication outside it. They are valid only when the base satisfies , , and every logarithmic argument is positive.
For base , where and , and positive values and :
| Law | Formula | Meaning |
|---|---|---|
| Product law | Multiplication becomes addition. | |
| Quotient law | ||
| Power law |
These laws follow directly from exponent laws. If and , then . Taking logarithms gives , which is the product law. Division uses subtraction of exponents, while a power multiplies an exponent, producing the quotient and power laws.
Simplify the expression:
Combine the product before applying the quotient:
The argument is positive, so the final logarithm is defined. The power law also works in reverse. For example, provided :
When solving equations, this allows an exponent to move in front of a logarithm or a coefficient to move back inside as an exponent.
A common misconception is that logarithms distribute over addition:
This is false. Logarithm laws split only multiplication, division, and powers; they do not split sums or differences. Another error is ignoring positivity. The original argument of every real-valued logarithm must remain positive, so check the domain before simplifying or solving.
This topic is AHL 1.9, so it is HL-only in Mathematics: Applications and Interpretation. In an exam, show the law used, preserve brackets, and state restrictions when variables are present. Keep exact logarithmic forms until the final step, then use the GDC if a numerical value is required. Examiners expect valid algebraic steps, not unsupported calculator output.
| Division becomes subtraction. |
| An exponent becomes a coefficient. |
