Log Calculator
Calculate natural logarithms (ln), common logarithms (log10), and logarithms of any base.
Logarithm Calculator
Must be greater than 0
Must be positive and ≠ 1
Common Applications
- Base 10: pH scale, decibels, Richter scale
- Base e: Exponential growth/decay, compound interest
- Base 2: Computer science, information theory, binary
Logarithm Rules
- log(xy) = log(x) + log(y)
- log(x/y) = log(x) - log(y)
- log(x^n) = n · log(x)
- log_b(b) = 1, log_b(1) = 0
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How to Use This Calculator
- Enter the number you want to find the logarithm of (must be positive)
- Select or enter the base: use quick buttons for common bases (2, e, 10) or enter a custom base
- Click Calculate to compute the logarithm and see results in multiple bases
- The calculator shows log with your chosen base, plus natural log (ln), common log (log₁₀), binary log (log₂), and the antilog
Formula
log_b(x) = ln(x) / ln(b) where ln is natural logarithm (base e). Common logarithms: log₁₀(x), ln(x) = log_e(x), log₂(x). Antilog: b^(log_b(x)) = x. Properties: log(xy) = log(x) + log(y), log(x/y) = log(x) - log(y), log(x^n) = n·log(x)
Frequently Asked Questions
What is a logarithm?▼
A logarithm answers the question: 'To what power must we raise the base to get this number?' For example, log₁₀(100) = 2 because 10² = 100.
What's the difference between ln and log?▼
ln (natural log) uses base e (≈2.71828), commonly used in calculus and science. log typically means log₁₀ (common log, base 10), often used in engineering and pH calculations. Both follow the same logarithm rules.
Why can't we take the log of zero or negative numbers?▼
No positive base raised to any power can equal zero or a negative number, so logarithms of non-positive numbers are undefined in real numbers (though they exist in complex numbers).
What is an antilog?▼
Antilog is the inverse operation of logarithm. If log_b(x) = y, then antilog_b(y) = x. It's equivalent to b^y. For example, if log₁₀(100) = 2, then antilog₁₀(2) = 100.