Logarithmic Functions

Calculus Readiness · Axiom Academy

The inverse of the exponential: a logarithm just answers the question " b to what power gives x ?" 1. A Logarithm Undoes an Exponential A logarithmic function has the form , where the base satisfies b > 0 and . It is the inverse of y = b^x : whatever the exponential builds up, the logarithm takes back down. Read it as a question: " b to what power gives x ?" The log equation and the exponential equation say the same thing 2. Evaluating Logs in Your Head The fastest way to read a log is to flip it into an exponent question. The animation climbs the powers of 2 — 2^0, 2^1, 2^2, 2^3, 2^4 landing on 1, 2, 4, 8, 16 — so to find you just climb until you hit 8 and read off the exponent: 3 . because 5^0 = 1 — the log of 1 is always 0 . because — fractions give negative logs. Two bases get their own notation Common log: , base 10 — used in pH, decibels, the Richter scale. Natural log: , base — everywhere in calculus and continuous growth. So and . Every log curve (for b > 1 ) has the same silhouette. The animation traces : as x shrinks toward 0 the curve plunges toward the vertical asymptote x = 0 , it crosses the axis at (1, 0) , and then it climbs — but agonizingly slowly. Domain : you can only take the log of a positive number — the curve never reaches or crosses the y -axis. Range: all real numbers. Grows very slowly: . Even a million has a tiny log — logs lag behind every polynomial. The curve dives without bound down the vertical asymptote x = 0 — but it never touches it.

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