What is the change of base formula?
log_b(a) equals ln(a)/ln(b) or log₁₀(a)/log₁₀(b).
Change of Base Calculator
Compute log_b(a) via ln(a)/ln(b) for positive a and positive b ≠ 1.
Enter argument and base
Provide a > 0 and base b > 0, b ≠ 1.
About this calculator
How to use Change of Base Calculator, plus the formulas and logic behind the results.
Tool description
Most calculators only offer ln or log₁₀. Change of base lets you evaluate any valid logarithmic base.
How to use
- Enter the argument a > 0.
- Enter the base b > 0 with b ≠ 1.
- Select Calculate to view log_b(a).
- Optionally compare with log₁₀ form.
Formulas & reference
log_b(a) = ln(a) / ln(b)log_b(a) = log₁₀(a) / log₁₀(b)
Examples
- log₂(8) = 3
- log₅(25) = 2
- log₁₀(1000) = 3
- Same result via ln or log₁₀ change of base
Calculation logic
Validates a > 0 and b > 0, b ≠ 1, then divides ln(a) by ln(b).
Base 1 is invalid because ln(1) = 0.
Change of Base Calculator FAQ
Common questions about this calculator and how to use the results.
Why can’t base be 1?
Every power of 1 is 1, so log₁ is undefined.
Can base be between 0 and 1?
Yes, if it is positive and not 1 (e.g. log_{0.5}(8) = −3).
Must a be positive?
Yes for real logarithms.
Do ln and log₁₀ versions match?
Yes, aside from floating-point rounding.
When is this used?
Whenever your calculator lacks the base you need.