Log Calculator
Logarithm of any number to any base — including base 10 (common log) and base e (natural log)
Frequently Asked Questions
A logarithm answers the question "what exponent do I need to raise this base to, to get this result" — the formal mathematical inverse of exponentiation — and this calculator handles logarithms to any base, including the two most commonly used: base 10 (common log) and base e (natural log).
Why logarithms and exponents are inverse operations
If raising a base to an exponent gives a result (base^exponent = result), then the logarithm reverses that relationship: log-base(result) = exponent — logarithms and exponents are two ways of expressing the exact same three-way relationship between base, exponent, and result, just solving for a different one of the three. Common logarithm (base 10) is widely used in contexts like the pH scale in chemistry and the decibel scale in acoustics, both of which measure quantities that span an enormous range and benefit from a logarithmic (rather than linear) scale to stay human-readable. Natural logarithm (base e, where e is a specific mathematical constant approximately 2.71828) shows up extensively in calculus, compound growth and decay calculations, and many natural scientific processes where continuous exponential change is involved.
A worked example
The same number's common logarithm (base 10) and natural logarithm (base e) are different values, since they answer "what power of 10" versus "what power of e" respectively — the two aren't interchangeable, and using the wrong base for a specific context (using natural log where a formula specifically calls for common log, or vice versa) produces an incorrect result even though both operations are conceptually "a logarithm."
How this connects to your other math calculations
Logarithms are the direct inverse of the operations the Exponent Calculator performs — for the specific case of finding what root a number represents (a related but distinct question from finding what logarithm it represents), the Root Calculator handles that adjacent calculation.
Common mistakes
Using the wrong logarithm base for a specific formula or context — applying natural log where common log was intended, or vice versa — produces a numerically different and incorrect result, since the two bases aren't interchangeable; always confirm which base a specific application or formula actually requires before calculating.