Compute log_b(x) for any base, solve for base or argument, work exponentials b^y, with change-of-base formula and reference table.
Compute log_b(x)
Solve for base: log_b(x) = y
Given x and y, finds b such that b^y = x
Solve for argument: log_b(x) = y
Given b and y, finds x = b^y
Exponential: b^y = ?
Raise base b to power y (inverse of logarithm)
Result
Change-of-base formula
Any logarithm can be expressed in any other base:
Most common: use natural log (ln) — log_b(x) = ln(x) / ln(b)
Common values reference
| x | log_2(x) | log_10(x) | ln(x) |
|---|
About logarithms
What is a logarithm?
A logarithm answers the question: to what power must a base be raised to produce a given number? If b^y = x, then log_b(x) = y.
Example: 2^3 = 8, so log_2(8) = 3.
Natural log (ln) vs common log (lg)
ln(x) is log with base e (~2.71828), used in calculus and continuous growth.
lg(x) or log(x) is log with base 10, used in engineering (pH, decibels, Richter scale).
log_2(x) is binary log, used in computer science and information theory.
Key properties
log_b(xy) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) - log_b(y)
log_b(x^n) = n * log_b(x)
log_b(1) = 0 and log_b(b) = 1
Domain restrictions
The argument x must be positive (x > 0). The base b must be positive and not equal to 1 (b > 0, b != 1).
Logarithms of zero or negative numbers are undefined in the real number system.
This logarithm calculator handles four tasks in one interface: computing log_b(x) for any base, solving for the base when argument and result are known, solving for the argument from base and result, and evaluating exponentials b^y. Select base 2, e (natural log), 10 (common log), 5, or enter any custom positive base other than 1. The tool shows the decimal value, flags integer results as exact, and prints the change-of-base formula ln(x)/ln(b) used in the calculation. Example: log_2(8) = 3 exactly, since 2^3 = 8. Example: ln(10) = 2.302585, since e^2.302585 = 10. A reference table lists log_2, log_10, and ln for common values (1, 2, 4, 8, 10, 16, 100, 1000). Domain rules are enforced: the argument must be positive, and the base must be positive and not equal to 1.