Exponentials and Logarithms
Exponentials and Logarithms
The exam tests the link between y = a^x and y = log_a x as inverse functions, the laws of logarithms, and equations where the unknown sits in an exponent. The natural base appears routinely: e^x and ln x turn up again in the calculus sections, so fluency here pays twice.
Core rules
- log_a x = y ⇔ a^y = x, with a > 0, a ≠ 1 and x > 0.
- Laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^n) = n log x; log_a a = 1; log_a 1 = 0.
- Change of base: log_a b = log_c b / log_c a, most often ln b / ln a.
- The graphs y = e^x and y = ln x are reflections in y = x; e^x > 0 for all x, and ln x → −∞ as x → 0⁺.
Worked example
Solve 3^(2x) − 10(3^x) + 9 = 0. Let y = 3^x. Then y² − 10y + 9 = 0, so (y − 1)(y − 9) = 0, giving y = 1 or y = 9. Since y = 3^x > 0, both are acceptable: 3^x = 1 gives x = 0, and 3^x = 9 = 3² gives x = 2. The substitution is the key step - without it the equation is not quadratic and cannot be factorised.
Traps
- Writing log(A + B) = log A + log B; the product law does not split a sum.
- Dropping the restriction x > 0 and offering the logarithm of a negative number as a solution.
- Taking logarithms of only one side of an equation.
- Forgetting that a hidden quadratic in e^x may produce a negative root, which must be rejected.