Logarithms and Exponentials

ACT Math · Axiom Academy

LESSON Logarithms and Exponentials Exponents and logarithms are two ways of reading the same relationship — master the flip between them and every log rule falls out of an exponent rule you already know. 1. Exponent Rules — What You Already Know An exponent just counts repeated factors: x^4 means . Every log rule in this lesson is really an exponent rule wearing a disguise, so a quick, solid review pays off twice. Product of powers: add the exponents Quotient of powers: subtract the exponents (x^2)^3 = x^6 — multiply the exponents. — the denominator is the root. 2. The Core Relationship — Flipping the Form A logarithm is the exponent an exponential is hiding. The two forms below say the exact same thing about the same three numbers — they just ask the question in a different order. b^x = y — "base b raised to x gives y ." x is the unknown you're solving FOR when you don't know the exponent. — "what power of b gives y ?" Same b , same x , same y — just isolated differently. 3. Common Log and Natural Log — Two Reserved Shortcuts Two bases come up so often that they get their own notation, with the base left off entirely. Common log — base 10, no base written Since we count in base 10, , , — the common log of a power of 10 is just that power. for the same reason ( 10^0 = 1 ), and this holds for too: and since e^1 = e . 4. Log Properties — The Exponent Rules, Translated

This is the written version of the interactive lesson above. See the full ACT Math course.