Evaluating Logarithms
Simple Explanation
To evaluate log_b(x) by hand, ask "what power of b gives x?" — often you can answer this directly by recognizing x as a power of b you already know.
Why Do We Need It?
Being able to evaluate simple logarithms mentally builds the number sense needed to check whether a calculator's answer (or an algebraic simplification) is reasonable.
Formula
Definition of the Logarithm
log_b(x) = y ⇔ bʸ = x (b > 0, b ≠ 1, x > 0)
A logarithm answers "to what power must the base b be raised to get x?" — it is the exponent y in the equivalent exponential equation bʸ = x.
- b
- — the base of the logarithm (b > 0, b ≠ 1)
- x
- — the argument — the number you are taking the logarithm of (x > 0)
- y
- — the logarithm itself — the exponent that produces x
When to use it: Whenever you need to find the exponent that produces a known result, or to switch between logarithmic and exponential form.
Worked Example
Evaluate a logarithm by recognizing a power
Evaluate log₄(64).
Why Does This Work?
By the definition of a logarithm, log_b(x) is exactly the exponent that turns b into x — so finding that exponent by trial (or by recognizing a familiar power) is a direct, valid way to evaluate it.
Real-Life Example
Reading sound intensity in decibels
Decibel level is calculated using log₁₀ of a sound intensity ratio.
Recognizing that an intensity ratio of 100 is 10², an audio engineer can immediately read off log₁₀(100) = 2, without a calculator.
Practice
Evaluate log₂(32).
EasyEvaluate log₅(1/25). (Give a negative number.)
MediumCommon mistake
Trying to evaluate log_b(x) when x is not a clean power of b, without switching to the change of base formula or a calculator — not every logarithm has a "nice" whole-number answer.
Quick Review
- log_b(x) equals the exponent that turns b into x.
- Evaluate by recognizing x as a familiar power of b.
- Negative results mean x is a fraction less than 1 (a negative power of b).