Exponential inequalities and the logarithm

The logarithm of b > 0 to the base a (a > 0, a ≠ 1) is the exponent to which a must be raised to get b: log_a b = c ⇔ aᶜ = b. For example log₂ 8 = 3 and log₃(1/9) = −2. Main properties: a^(log_a b) = b; log_a 1 = 0; log_a a = 1; log_a(xy) = log_a x + log_a y; log_a(x/y) = log_a x − log_a y; log_a xⁿ = n log_a x (x, y > 0). lg x is the logarithm to base 10 and ln x is the logarithm to base e ≈ 2.718. The function y = log_a x has domain (0, +∞), range (−∞, +∞) and zero x = 1; it is increasing for a > 1 and decreasing for 0 < a < 1. The simplest equation log_a x = b has the solution x = aᵇ; log_a f = log_a g gives f = g, but we must check f > 0 and g > 0. In an inequality the sign is kept for a > 1 and reversed for 0 < a < 1, and the expression under the logarithm must always be positive.