Properties of Logarithms
Grade 10 Algebra — Master the fundamental rules that make logarithms powerful
Main Theorem — The Three Fundamental Properties
Conditions: Let $a$, $M$, and $N$ be positive numbers with $a \neq 1$, and let $\gamma$ be any real number. Then the following three properties hold:
Property 1: Product Rule
$$\log_a(M \cdot N) = \log_a M + \log_a N$$
The logarithm of a product of positive numbers equals the sum of their logarithms.
Property 2: Quotient Rule
$$\log_a\!\left(\frac{M}{N}\right) = \log_a M - \log_a N$$
The logarithm of a quotient of positive numbers equals the difference of the logarithms of dividend and divisor.
Property 3: Power Rule
$$\log_a(M^{\gamma}) = \gamma \cdot \log_a M$$
The logarithm of a power of a positive number equals the exponent multiplied by the logarithm of that number.
Visual Overview: How the Properties Connect
Interactive visualization — available in the full lesson