Free Logarithm Calculator
Logarithm in any base, plus natural log and log base 10.
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Inputs
Result
log_10(1000)
3.00000000
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Short answer
log_b(x) = ln(x) / ln(b). The logarithm is the exponent to which the base must be raised to produce x.
What is the Log Calculator?
The logarithm is the inverse of exponentiation. log_b(x) answers 'what power of b gives x?'
How does the Log Calculator work?
Compute ln(x) and ln(b), then divide.
Formula
Variables
bBase of the logarithmxPositive valuelnNatural log (base e)
Explanation
The change-of-base formula lets any calculator with only ln compute logs in any base.
Examples
Example 1: log₁₀(1000)
3 — because 10^3 = 1000.
Applications
- Decibels
- pH
- Richter scale
- Information theory (log₂)
Advantages
- Any base, plus ln, log₁₀, log₂ automatically
Limitations
- Requires strictly positive x and base ≠ 1
Common mistakes
- Trying to take log of zero or a negative number
Tips
- log(a·b) = log(a) + log(b) — logs turn multiplication into addition
Related concepts
The Log Calculator sits inside the Math Calculators hub, in the algebra & equations cluster. Solving for unknowns and manipulating expressions. Understanding the terms below makes the output easier to interpret and easier to compare against neighbouring measures.
Practical use cases and industry applications
Understanding Log
The logarithm is the inverse of exponentiation. log_b(x) answers 'what power of b gives x?' Within math calculators, log belongs to the algebra & equations cluster, where it shares terminology and assumptions with closely related tools.
Learning how log is calculated
The change-of-base formula lets any calculator with only ln compute logs in any base. Working through the variables one at a time — b, x, ln — makes the result reproducible by hand and easier to sanity-check.
Using log to make a decision
Decibels pH Richter scale Because outputs depend on the assumptions you enter, run more than one scenario before committing to a figure.
How log compares with related measures
equation, root, exponent, logarithm all describe adjacent aspects of algebra & equations. Comparing this calculator's output against those measures — using the related tools listed on this page — prevents a single metric from being read in isolation.
Frequently asked questions
What is ln vs log?
ln is base e (≈2.718). log with no base written is base 10 in most scientific contexts, base e in math papers.
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