Exponential and Logarithmic Functions
What you must understand
Exponential functions have the form f(x) = a*b^x with b positive and not equal to 1; they model growth (b > 1) and decay (0 < b < 1). Logarithms are inverses of exponentials: y = log_b(x) means b^y = x. Master the log laws: the product, quotient, and power rules, plus the change-of-base formula log_b(x) = ln x / ln b. To solve an exponential equation, isolate the exponential and take a logarithm of both sides; to solve a logarithmic equation, condense to a single log and rewrite in exponential form. Domains matter: logarithms require a positive argument, and exponentials are always positive. Graphs of exponentials have a horizontal asymptote, while log graphs have a vertical asymptote.
Traps to watch
- Writing log(a + b) = log a + log b, which is invalid.
- Forgetting that log_b(x) is undefined for x <= 0.
- Dropping a solution that makes a log argument negative or zero.
- Applying the power rule backwards: log(x^2) is 2log|x|, not 2log x.
- Confusing the horizontal asymptote of an exponential with the vertical asymptote of a logarithm.
Worked example
Solve 3^(2x) = 81. Write 81 as 3^4, so 3^(2x) = 3^4, giving 2x = 4 and x = 2. Alternatively, take log base 3 of both sides: 2x = log_3(81) = 4.
Rule of thumb
Rewrite both sides with the same base when possible; otherwise take a log, and always check that log arguments stay positive.