Logarithms and the Change of Base Formula

Calculates the logarithm to the given base as log_b x = ln x ÷ ln b, the power to which b must be raised to give x. The common logarithm (base 10) and the natural logarithm (base e) are also shown.

Explanation

A logarithm answers one question: to what power must the base be raised to give this number? Writing $\log_b x = y$ says exactly the same thing as $b^y = x$ . The logarithm undoes the power.

\log_b x = y \iff b^y = x

A logarithm to any base can be rebuilt from natural logarithms through the change of base formula, which is what this calculator does internally.

\log_b x = \dfrac{\ln x}{\ln b}

$x$ — the value, which must be positive
$b$ — the base, positive and not equal to 1
$\log_{10} x$ — the common logarithm, base 10
$\ln x$ — the natural logarithm, base $e \approx 2.71828$

Example

Take $x = 8$ with base $b = 2$ . Since 2 cubed is 8,

\log_2 8 = 3

The change of base formula agrees: $\dfrac{\ln 8}{\ln 2} = \dfrac{2.0794}{0.6931} = 3$ . The calculator also shows $\log_{10} 8 \approx 0.9031$ and $\ln 8 \approx 2.0794$ .

Notes

The value $x$ must be positive. No power of a positive base ever reaches zero or turns negative, so the logarithm of such a number is not a real number, and the calculator rejects it.
The base must be positive and not 1. Every power of 1 is 1, so base 1 could never produce anything else.
$\log_b 1 = 0$ and $\log_b b = 1$ , whatever the base happens to be.
For $0 < x < 1$ with a base above 1, the logarithm is negative: $\log_{10} 0.01 = -2$ .
Logarithms turn multiplication into addition, since $\log_b (xy) = \log_b x + \log_b y$ . That property is what made them indispensable before calculators existed.