Logarithm Calculator – Calculate Log Values
Calculate logarithms with any base.
Table of Contents
How to Use
- Enter the number you want to find the logarithm of
- Select the logarithm type (log₁₀, ln, log₂, or custom)
- For custom base, enter your desired base value
- Click calculate to see the result
What is a Logarithm?
A logarithm answers the question: 'To what power must we raise the base to get a certain number?' If b^x = y, then log_b(y) = x. The logarithm is the inverse operation of exponentiation.
For example, log₁₀(100) = 2 because 10² = 100, and ln(e) = 1 because e¹ = e.
Types of Logarithms
| Type | Base | Notation | Common Use |
|---|---|---|---|
| Common Log | 10 | log(x) or log₁₀(x) | Scientific calculations, pH scale |
| Natural Log | e ≈ 2.718 | ln(x) | Calculus, growth/decay models |
| Binary Log | 2 | log₂(x) | Computer science, information theory |
| Custom | Any b > 0, b ≠ 1 | log_b(x) | Specialized applications |
Key Logarithm Properties
- log_b(xy) = log_b(x) + log_b(y) — Product rule
- log_b(x/y) = log_b(x) - log_b(y) — Quotient rule
- log_b(x^n) = n · log_b(x) — Power rule
- log_b(b) = 1 — Logarithm of the base
- log_b(1) = 0 — Logarithm of 1
- Change of base: log_b(x) = ln(x) / ln(b)
Frequently Asked Questions
- Why can't I take the logarithm of a negative number or zero?
- In real numbers, logarithms are only defined for positive values. There's no real power you can raise a positive base to and get a negative number or zero. Complex logarithms exist but require complex number theory.
- What's the difference between log and ln?
- log (or log₁₀) uses base 10 and is common in science and engineering. ln uses base e (≈2.718) and is essential in calculus because the derivative of ln(x) is simply 1/x.
- Why is base 1 not allowed?
- 1 raised to any power always equals 1, so there's no unique exponent that gives other values. This makes log₁(x) undefined for any x ≠ 1.