What are the laws of logarithms?

The three core logarithmic rules

The laws of logarithms stem directly from exponent rules because a logarithm represents an exponent. When multiplying like bases, you add exponents. Therefore, the logarithm of a product is the sum of the individual logarithms: log_b(xy) = log_b(x) + log_b(y). For example, log_2(8 * 4) = log_2(8) + log_2(4) = 3 + 2 = 5.

Similarly, dividing like bases means subtracting exponents. The quotient rule states that log_b(x / y) = log_b(x) - log_b(y). Raising a power to another power multiplies exponents, which gives the power rule: log_b(x^k) = k * log_b(x). This rule pulls exponents out front as multipliers, turning complex powers into straightforward products.

Expanding, condensing, and changing base

Expanding means breaking a single compact logarithm into several smaller parts. For instance, log_3(x^2 / y) expands into 2 * log_3(x) - log_3(y). Condensing works in reverse: combine a sum or difference of logarithmic terms that share the same base back into a single term. To condense 3 * log_10(a) + log_10(b), write it as log_10(a^3 * b).

Calculators usually calculate only common logs (base 10) or natural logs (base e). To evaluate an arbitrary base like log_3(7), apply the change-of-base formula: log_b(x) = log_c(x) / log_c(b). Computing log(7) / log(3) on a standard calculator yields approximately 1.771.

The most common trap to avoid

A frequent error in 11th grade algebra is treating logarithms as distributive multipliers across sums. The expression log_b(x + y) does not equal log_b(x) + log_b(y). Logarithms split products into sums, never sums into sums. If you see log(x + 5), leave the argument inside the parentheses as a single term unless other factoring applies.