Change Of Base Logarithm Calculator

Work out change of base logarithm instantly with clear inputs, formula shown and shareable results.

About Change Of Base Logarithm Calculator

Calculators usually offer only ln and log₁₀, so any other base is reached by the change of base rule log_b x = ln x / ln b. The same ratio works with logarithms to any common base.

Formula

Change of base

log_b x = ln x / ln b = log₁₀ x / log₁₀ b

Worked example

With the default values below, Change Of Base Logarithm Calculator returns log_b x of 2.86135312. Change any input above and every figure here is recalculated in your browser.

Inputs used

Value xminimum 0: 100
Base bminimum 0: 5

Results

log_b x: 2.86135312
ln x: 4.60517019
ln b: 1.60943791
b raised to the answer (should be x): 100.000000

Step by step

1Values used
2Change of base
3log_b x= 2.86135312
4ln x= 4.60517019
5ln b= 1.60943791
6b raised to the answer (should be x)= 100.000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is log₅ 100?

ln 100 / ln 5 ≈ 4.6051702 / 1.6094379 ≈ 2.8613531.

Why is base 1 forbidden?

Because ln 1 = 0, so the formula divides by zero — and 1 raised to any power is always 1, so the logarithm cannot be defined.