Calculate logarithms with any arbitrary positive base (log_b x) using the change of base formula with step-by-step mathematical steps.
Did this calculation save you time?
Student ProjectHi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.
I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!
A logarithm is the inverse operation to exponentiation. The logarithm of a number x with respect to base b is the exponent to which b must be raised to produce x: log_b(x) = y if and only if b^y = x.
\log_b(x) = y \iff b^y = x, \quad \log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}To compute a logarithm to any base on standard computing hardware, divide the natural logarithm of x by the natural logarithm of base b.
Scenario: Calculate log₂(32)
Given: argument x = 32, base b = 2
By change of base: log₂(32) = ln(32) / ln(2)
Notice that 2⁵ = 32
Result: log₂(32) = 5
The change of base formula allows you to evaluate any base b logarithm using natural log or common log: log_b(x) = ln(x) / ln(b).
Because 1 raised to any power is always 1 (1^y = 1). Therefore, you cannot solve 1^y = x for any number other than 1.
In the real number system, raising a positive base b to any real power always yields a positive number (b^y > 0). Negative arguments require complex logarithms.
For any valid base b, log_b(1) = 0 because b⁰ = 1.
Calculate the natural logarithm (ln x = log_e x) with base e ≈ 2.718281828 with step-by-step calculus and exponential properties.
Calculate common base-10 logarithms (log₁₀ x) with characteristic, mantissa, and scientific notation connections.
Calculate any base raised to the power of n (x^n) for positive, negative, and fractional exponents with step-by-step exponent laws.