Evaluating Logarithms

Pre-Calculus · Axiom Academy

A logarithm is just an exponent in disguise. Learn to read it that way and you can evaluate logs in your head — no calculator. The log form and the exponential form say the same thing two ways . The base b stays the base in both. The answer of the log — the value y — is exactly the exponent on the other side. The number x is what that power produces. The log's answer y is the exponent b raised to that exponent gives x 2. Find the Power by Searching To evaluate , ask " 2 to what power gives 8 ?" and just walk up the powers of 2 until one lands on 8 . The exponent that hits the target is your answer. : count 3, 9, 27, 81 — that is four steps, so . : count 10, 100, 1000 — three steps, so . 3. Three Values You Never Have to Compute Some logs follow instantly from "a log is an exponent." Whatever the base b is, these three are always true — because b^0 = 1 , b^1 = b , and b^k literally carries its exponent. Any base to the power 0 is 1 , so the exponent that makes 1 is always 0 . Any base to the power 1 is itself, so the exponent that makes b is 1 . When the number already shows the power, just read off the exponent. A log undoes a power: ask " b to what power gives b^k ?" and the answer is right there — k . 4. When the Answer Is Negative If x is a fraction between 0 and 1 , the exponent goes negative . To evaluate , step the powers of 5 downward through 5^0, 5^ -1 , 5^ -2 until you reach . The log is positive — you climb above b^0 = 1 .

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