Log Calculator
Find the logarithm of a number in any base, with the natural log, log base 10 and log base 2.
Results
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Steps
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Quick Answer
A logarithm answers the question: what power of the base gives this number? Log base 10 of 1,000 is 3, because 10³ = 1,000. To find a log in any base, divide natural logs: log b (x) = ln(x) ÷ ln(b).
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Table of Contents
How This Log Calculator Works
Enter a number and a base. The calculator finds the logarithm in that base, and also shows the natural log, log base 10 and log base 2, so you can compare them.
Quick explanation: if b^y = x, then log b (x) = y.
- Base 10: the common logarithm.
- Base e: the natural logarithm, written ln.
- Base 2: often used in computing.
Core Formulas
- Change of base: log b (x) = ln(x) ÷ ln(b)
- Product rule: log(xy) = log(x) + log(y)
- Quotient rule: log(x ÷ y) = log(x) − log(y)
- Power rule: log(x^k) = k × log(x)
Worked Example
Find log base 2 of 32.
- ln(32) = 3.465736
- ln(2) = 0.693147
- 3.465736 ÷ 0.693147 = 5
- Check: 2⁵ = 32
Common Mistakes
- Log of zero or a negative: this is not defined for real numbers.
- Base of 1: this is not allowed, because 1 to any power is 1.
- Mixing up log and ln: log usually means base 10, and ln means base e.
Frequently Asked Questions
It is the power you raise a base to in order to get a number. Log base 10 of 100 is 2, because 10² = 100.
Log often means base 10, and ln means base e, which is about 2.718.
Divide the natural log of the number by the natural log of the base.
It is 0 in any allowed base, because any such base to the power 0 is 1.
Not with real numbers. The input must be greater than zero.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Logarithm | Definition, rules and change of base. | https://en.wikipedia.org/wiki/Logarithm |