Log Calculator
Calculate logarithm log_b(x) with any base.
Asking what exponent produced a number
A logarithm answers the inverse question to a power. If 10³ = 1000, thenlog₁₀(1000) = 3. In words: the logarithm is the exponent the base needs to reach the number. Three bases dominate — base 10 (written log), base e(written ln), and base 2, which appears throughout computing.
The properties are what made logarithms historically indispensable, and they still matter:log(ab) = log a + log b, log(a/b) = log a − log b, andlog(aⁿ) = n log a. Multiplication becomes addition and powers become multiplication, which is exactly why slide rules worked, and why the last rule is the standard way to get an unknown out of an exponent when solving equations.
Two constraints follow from the definition. You cannot take the logarithm of zero or a negative number in the reals, because no real exponent makes a positive base produce them. And any base can be converted to any other by log_b(x) = ln(x) / ln(b), which is how a calculator offering only log and ln still handles base 7. Logarithmic scales — decibels, pH, Richter — exist because they compress enormous ranges into readable numbers.
Frequently Asked Questions
What is a logarithm?
The logarithm log_b(x) is the exponent to which the base b must be raised to produce x.
What is the change of base formula?
log_b(x) = ln(x) / ln(b) = log10(x) / log10(b).
What is a logarithm?
The inverse of exponentiation: log base b of x is the power you must raise b to in order to get x. Since 10^3 = 1000, log10(1000) = 3. Logarithms convert multiplication into addition, which is why they underpinned slide rules and why they linearise exponential growth on a chart.
What is the difference between log, ln and log2?
log usually means base 10, common in engineering and in scales like decibels and pH. ln is the natural log, base e, which arises wherever continuous growth or decay is modelled. log2 is base 2, standard in computing and information theory. Convert between any of them with log_b(x) = ln(x) / ln(b).