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Logarithm Explorer

Calculate logarithms in any base, solve log equations, and use the change-of-base formula. The graph shows both the logarithmic curve and its exponential inverse.

Graph

(8.00, 3.00)y = log_2(x)y = 2^x

Parameters

Results

log2(8)
3.00000

Log Properties

log(ab) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^n) = n · log(a)
log_b(1) = 0
log_b(b) = 1

Step-by-Step Calculation

1. Definition

log⁡2(8)=y  ⟺  2y=8\log_{2}(8) = y \iff 2^y = 8

2. Using change of base formula

log⁡2(8)=ln⁡(8)ln⁡(2)=2.079440.693147\log_{2}(8) = \frac{\ln(8)}{\ln(2)} = \frac{2.07944}{0.693147}

3. Result

log⁡2(8)=3.00000\log_{2}(8) = 3.00000

Reference Guide

Definition

A logarithm answers the question "what exponent gives this result?" It is the inverse of exponentiation.

log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

Change of Base

You can convert a logarithm to any other base using this formula.

log⁡b(x)=log⁡c(x)log⁡c(b)\log_b(x) = \frac{\log_c(x)}{\log_c(b)}

Log Properties

log⁡(ab)=log⁡(a)+log⁡(b)\log(ab) = \log(a) + \log(b)
log⁡(a/b)=log⁡(a)−log⁡(b)\log(a/b) = \log(a) - \log(b)
log⁡(an)=n⋅log⁡(a)\log(a^n) = n \cdot \log(a)

Natural Logarithm

The natural log (ln) uses base e, where e is approximately 2.71828. It appears throughout calculus, physics, and probability because e has special properties related to continuous growth.