Exponential and Logarithmic Functions
What the exam tests
Exponential and Logarithmic Functions tests laws of exponents and logarithms, domains, monotonicity, and equations that require checking for extraneous solutions. Kyoto problems often use a substitution such as t = 2^x or ask for the range of a log expression.
Core rules
- For a > 0, a ≠ 1, the function a^x is defined for all real x. It is increasing if a > 1 and decreasing if 0 < a < 1.
- log_a x is defined only for x > 0, with a > 0 and a ≠ 1. It is the inverse of a^x.
- Laws: log_a(MN) = log_a M + log_a N and log_a(M/N) = log_a M - log_a N, valid only when M, N > 0. Also log_a(M^r) = r log_a M.
- Change of base: log_a b = log_c b / log_c a.
- When solving logarithmic equations, first state the domain, then check every candidate in the original equation.
Worked example
Solve log_2(x - 1) + log_2(x + 3) = 3. The domain requires x - 1 > 0 and x + 3 > 0, so x > 1. Combine: log_2[(x - 1)(x + 3)] = 3, hence (x - 1)(x + 3) = 8. Expanding gives x^2 + 2x - 3 = 8, so x^2 + 2x - 11 = 0. The roots are x = -1 ± 2√3. Only -1 + 2√3 is greater than 1, so it is the only solution.
Traps
Do not write log(M + N) = log M + log N. Do not ignore the domain after combining logs. In exponential equations, do not assume a^x = a^y implies x = y unless the base is positive and not 1. Check whether a substitution creates a negative value for an exponential expression.