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

Parameters

Mode

Base (b) Use 2.718 for e

Value (x)

Presets

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 \iff 2^y = 8

2. Using change of base formula

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

3. Result

\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 \iff b^y = x

Change of Base

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

\log_b(x) = \frac{\log_c(x)}{\log_c(b)}

Log Properties

\log(ab) = \log(a) + \log(b)

\log(a/b) = \log(a) - \log(b)

\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.

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