Logarithms Summary

Pre-Calculus · Axiom Academy

Everything from the logarithms unit in one place: the definition, the properties, change of base, and how to solve exponential and logarithmic equations. A logarithm answers one question: to what power must the base be raised to get this number? is just b^y=x written the other way. Logs and exponentials are inverses, so they undo each other: and . The properties turn products into sums, quotients into differences, and exponents into coefficients — that is what makes logs useful for solving equations. To solve b^x = N , take a log of both sides. To solve a log equation, rewrite it in exponential form — then check every answer, because the argument of a log must stay positive. Core Concept Definition of a Logarithm The logarithm base b of x is the exponent you put on b to produce x . The two equations say exactly the same thing — one is the log form, the other the exponential form. Core Concept Common & Natural Logs Two bases come up so often they get their own shorthand. The common log is base 10 (written with no base), and the natural log is base e (written ). Core Concept Converting & Evaluating To evaluate a log by hand, switch to exponential form and ask which exponent works. For perfect powers this is exact; otherwise use change of base and a calculator. To convert: becomes b^ y =x and vice versa. Watch out for: for every valid base.

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