Logarithm Calculator
Find the exponent needed to raise a chosen logarithm base to a positive number.
Logarithm
Real logarithms only: x > 0, b > 0 and b ≠ 1. Floating-point results are rounded for display. Bases extremely close to 1 can magnify small input differences.
Formula checked · NumberSwift · Review process
How this calculator works
The logarithm answers “what exponent turns this base into the entered number?” Change of base divides the natural logarithm of the number by the natural logarithm of the base. A base between zero and one is valid and reverses the growth direction. The number must be positive, and base one cannot define a unique exponent. The natural-base preset uses the mathematical constant e rather than a rounded user value.
Formula
log_b(x) = ln(x)/ln(b), so b^(log_b(x)) = x. Natural log uses b=e; common log uses b=10.
Assumptions & limitations
Real logarithms only: x > 0, b > 0 and b ≠ 1. Floating-point results are rounded for display. Bases extremely close to 1 can magnify small input differences.
Worked example
log_10(100) = 2 because 10² = 100. log_2(8) = 3. ln(e) = 1. With custom base 0.5, log_0.5(4) = −2.
Our approach to calculations ↗Common questions
Why can the logarithm be negative?
With a base above one, positive numbers below one need negative exponents. Bases below one reverse that relationship.
Why is base one invalid?
One raised to any real power remains one, so it cannot give a unique inverse for general numbers.
Can I take the log of zero or a negative number?
Not within this real-number tool. Zero has no finite real logarithm and negative inputs require a different complex-number treatment.