Exponential and Logarithmic Expressions and Equations
Core concepts
Master exponent laws, the definition log_a b = c ⇔ a^c = b, logarithm laws (product, quotient, power, change of base), and the domains a > 0, a ≠ 1, b > 0. Solve exponential equations by equating exponents or taking logarithms, and logarithmic equations by combining logs and converting to exponential form. Know that a^x and log_a x are inverse functions.
Traps
- Extraneous solutions from combining logs: log(x − 1) + log(x + 2) = 1 requires x − 1 > 0 and x + 2 > 0 before solving.
- Forgetting that log_a 1 = 0 and log_a a = 1, and misapplying log(A + B) ≠ log A + log B.
- Ignoring that a^x > 0 for all real x, so a^x = negative has no solution.
- Misusing change of base in the wrong direction.
Worked example
Solve log₂(x − 1) + log₂(x + 2) = 2. Combine: log₂[(x − 1)(x + 2)] = 2, so (x − 1)(x + 2) = 4. Expanding: x² + x − 2 = 4, so x² + x − 6 = 0, giving (x + 3)(x − 2) = 0, so x = −3 or x = 2. Domain requires x > 1, so x = −3 is rejected. The only solution is x = 2.
Rule of thumb
Always determine the domain restrictions before solving a logarithmic equation; check each candidate against the original arguments.