When you know the base and the result but not the exponent, logarithms give the answer.
About this calculator
When you know the base and the result but not the exponent, logarithms give the answer. If 2^x = 32, then x = log₂(32) = 5 because 2⁵ = 32.
The formula x = ln(r) / ln(b) works for any valid base. Base must be positive and not equal to 1. Result must be positive for a real exponent.
This is the inverse of [[fractional-exponents]] and general [[exponent]] evaluation. For growth problems see [[half-life]] or compound interest tools. For log base 10 or e try [[log]].
Common uses
- Find x when 2^x = 32
- Solve exponential equations in algebra
- Determine how many doublings reach a target
- Check logarithm homework answers
- Convert between multiplicative growth steps
Frequently asked questions
What exponent x satisfies 2^x = 32?
x = 5 because 2⁵ = 32. Using logs: x = ln(32) / ln(2) ≈ 5.
Why must the base be positive and not 1?
b ≤ 0 or b = 1 breaks the log definition. b = 1 gives 1^x = 1 always; b ≤ 0 is not valid for real logs over all x.
Why must the result be positive?
A positive base raised to any real exponent stays positive. Negative or zero results have no real exponent solution.
How is this related to logarithms?
x = log_b(r) by definition. The calculator uses natural logs: x = ln(r) / ln(b).