Solving Logarithmic Equations Step by Step

Pre-Calculus · Axiom Academy

EXAMPLE Solving Logarithmic Equations Condense, convert to exponential form, solve — then check the domain and reject extraneous roots. Solve the logarithmic equation , where denotes the common (base 10 ) logarithm. Be sure to check each candidate against the domain. Nicely done. You condensed a sum of logs, switched to exponential form, solved the quadratic, and — crucially — threw out the root that broke the domain. Condense first: turns a sum of logs into a single log you can undo. Then undo the log: means A = 10^ 1 = 10 , which gives an ordinary quadratic. Always check the domain: a log is only defined for a positive argument, so any candidate that makes an argument is extraneous and must be rejected. Result: x = 5 is the only solution; x = -2 is extraneous because is undefined. Solving the algebra is only half the job — verifying against the domain is what makes the answer correct.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.