What Are the Laws of Logarithms (Product, Quotient, and Power Rules)?

The three laws of logarithms are the product rule, quotient rule, and power rule. They transform multiplication, division, and powers inside a logarithm into addition, subtraction, and multiplication outside it.

For base $a$, the conditions are $a\gt 0$, $a\neq 1$, and every logarithm argument must be positive. These restrictions matter because logarithms of zero or negative real numbers are undefined.

These laws follow from the laws of exponents. If $\log_a x=p$ and $\log_a y=q$, then and . Therefore,

$$xy=a^p a^q=a^{p+q},$$

so $\log_a(xy)=p+q=\log_a x+\log_a y$. Similarly, division uses , while powers use .

For example, assuming $x\gt 0$ and $y\gt 0$,

$$\log_2\left(\frac{8x^3}{y^2}\right)=\log_2 8+\log_2(x^3)-\log_2(y^2)=3+3\log_2 x-2\log_2 y.$$

Before applying a law, identify the operation immediately inside the logarithm. A product allows the product rule, a fraction allows the quotient rule, and an exponent allows the power rule. Apply the power rule before combining terms when this makes coefficients clearer. For instance, coefficients attached to logarithms become exponents when condensing, while subtraction indicates that the corresponding factor belongs in the denominator.

When solving an equation, first simplify each side with these laws, then isolate the logarithm or convert to exponential form. Check every candidate in the original equation, not only in the simplified equation.

The laws can also be applied in reverse to condense expressions. For example,

$$2\ln x-\ln y=\ln(x^2)-\ln y=\ln\left(\frac{x^2}{y}\right).$$

The same rules apply to common logarithms, $\log x$, and natural logarithms, $\ln x$.

A common misconception is that logarithms distribute over addition. This is false: $\log_a(x+y)\neq\log_a x+\log_a y$. There is no logarithm law for separating a sum or difference inside one logarithm.

Exam technique: This is IB Mathematics AA SL 1.7: Laws of Exponents and Logarithms, examinable on Papers 1 and 2. Show each transformation, preserve the subtraction sign in the quotient rule, and check that every logarithm argument is positive, especially when rejecting invalid solutions to logarithmic equations.