Methodology
How this calculator works
The calculator uses the change-of-base identity log_b(x) = ln(x) ÷ ln(b). It also evaluates ln(x), log₁₀(x), log₂(x), and raises the chosen base to the calculated result as a numerical verification.
Formula testedAlgebra & Equations
Logarithm Calculator
Find a logarithm in any valid base and compare it with natural, common and binary logarithms plus an inverse power check.
Algebra & equations
Find a logarithm in any valid base
Calculate log base b of x and compare the same positive value in natural, common and binary logarithms.
Your entries stay in this browser and are used only for this calculation.
Logarithm result
Enter your values
The logarithm and change-of-base checks will appear here.
Worked example
Reading the result
For x = 8 and base 2, ln(8) ÷ ln(2) = 3. The inverse check is 2³ = 8, confirming the result.
Inverse relationship
A logarithm answers an exponent question
Instead of asking for b raised to y, a logarithm starts with b and x and solves for y. The inverse check shown beside the answer makes that relationship visible.
Academic reference
Logarithmic functions
OpenStax: logarithmic functionsRelated algebra tools
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Common questions
Frequently asked questions
What does log base b of x mean?+
It is the exponent y that makes bʸ = x. For example, log₂(8) = 3 because 2³ = 8.
What is the difference between ln and log?+
Ln normally means base e. Common log means base 10, although notation can vary by subject and calculator.
How does the change-of-base formula work?+
Log base b of x equals ln(x) divided by ln(b), or equivalently log10(x) divided by log10(b).
Why must the input value be positive?+
A real logarithm is defined only for a positive argument. Zero and negative inputs require concepts outside the real logarithm domain.
Why can the base not equal 1?+
Every power of 1 remains 1, so it cannot generate a one-to-one exponential function with an inverse logarithm.
Can the base be between zero and one?+
Yes. Such a logarithm is valid and decreasing, so values greater than one produce negative logarithms.
Important scope
The real logarithm domain requires x > 0, b > 0 and b ≠ 1. Displayed decimals are rounded, while the inverse check uses the unrounded calculation. This tool is for numerical learning and verification; preserve exact logarithmic forms when required.