Free · Exponents and logarithms calculators 🇪

Real Logarithm with Any Valid Base

Calculate a real logarithm, recover its positive argument, or solve for a valid logarithm base.

  • Formula & worked example
  • Private in your browser
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

The logarithm of 32 in base 2 is 5 because 2⁵ = 32. A base of 0.5 is also valid: log₀.₅(8) = −3.

log_b(x) = ln(x)/ln(b); x = b^y

Worked example

Enter these known values and leave the other values blank.

Logarithm base (b)
2
Positive argument (x)
32
Logarithm value (y)
5

Assumptions and limitations

  • The real-valued argument must be greater than zero. The base must be positive and different from 1.
  • Bases between 0 and 1 are allowed. Their logarithms reverse the increasing/decreasing relationship seen with bases greater than 1.
  • Every valid base has log_b(1) = 0. Supplying only x = 1 and y = 0 does not uniquely determine the base.
  • Decimal results have finite precision. Integer-power examples can be exact, while many other logarithms are irrational.

Common questions

Why can’t the base be 1?

Every real power of 1 equals 1, so exponentiation with base 1 cannot be inverted into a logarithm on positive arguments.

Are logarithms of negative arguments supported?

This calculator works with real logarithms. Zero and negative arguments are outside that domain; complex logarithms require a different treatment.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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