Logarithm Calculator
Evaluate log_b(a), the natural log ln a, or the change-of-base formula ln a / ln b. Each run checks the answer by raising the base to the computed power.
log_2(8) = 3
- log_b(a) is the exponent that satisfies b^x = a.
- Change of base: ln 8 / ln 2 = 3.
- Check: 2^3 = 8.
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How to Use the Logarithm Calculator
Choose log_b(a), ln a, or Change of base. Enter a positive argument. For a custom base, enter a positive number other than 1. The check metric raises the base to the answer so you can see b^x return a.
- Select the form you want.
- Enter the argument, and the base unless you chose ln.
- Read the value, the change-of-base quotient, and the exponential check.
Evaluating b^n belongs on the Exponent Calculator — that page is the inverse of this one. Writing a number as a × 10^n belongs on the Scientific Notation Calculator. Solving A = P(1+r)^t for t belongs on the Compound Interest Calculator, which uses this logarithm.
What a Logarithm Is
A logarithm asks “what power of the base gives the argument?” 2³ = 8, so log₂(8) = 3. The natural log uses base e ≈ 2.71828. Common log uses base 10, which is why log₁₀(1000) = 3 matches the number of times you multiply 10 to reach 1000.
Logs of 0 or negatives are not real. Base 1 is unusable because 1^x is always 1. Base between 0 and 1 is allowed and reverses inequalities, but the identity b^x = a still holds.
Logarithm Calculator Examples
log₂(8) = 3
2 × 2 × 2 = 8, so the exponent is 3. The check metric shows 2³ = 8.
log₁₀(1000) = 3
10³ = 1000. Change of base: ln 1000 / ln 10 = 3.
ln e = 1
By definition e¹ = e, so ln e = 1.
log₂(1/8) = −3
2^(−3) = 1/8. A reciprocal argument produces a negative log.
log₁₀(1) = 0
Any allowed base to the power 0 is 1, so log_b(1) = 0.
log(0) is rejected
There is no real power of 10 that equals 0. The same is true of any negative argument.
Change of Base
Calculators compute ln (or log₁₀) natively. Any other base is a ratio: log_b(a) = ln a / ln b. That is the same identity used to solve A = P(1+r)^t for t: t = ln(A/P) / ln(1+r). The Compound Interest Calculator’s Find time mode is this formula with money labels.
Common Logarithm Mistakes
Writing log(a + b) as log a + log b
The sum rule is log(ab) = log a + log b. Addition inside the argument does not split.
Using base 1
1^x is always 1, so it cannot hit any other argument. The calculator rejects base 1.
Asking for log of a negative
No real power of a positive base is negative. Complex logs are out of scope here.
Confusing ln with log₁₀
ln is base e. Common log is base 10. They differ by the constant ln 10 ≈ 2.302585.
Treating this as an exponent evaluator
This page finds x in b^x = a. To compute b^x when x is known, use the Exponent Calculator.
Logarithm Calculator FAQs
What is a logarithm?
log_b(a) is the exponent x such that b^x = a. log₂(8) = 3 because 2³ = 8.
What is the change-of-base formula?
log_b(a) = ln a / ln b (or log₁₀ a / log₁₀ b). Any allowed base can be the intermediate.
Why is log of 0 undefined?
A positive base raised to a real power is never 0. The values approach 0 only as the exponent goes to −∞.
Is this the inverse of the Exponent Calculator?
Yes. Exponent evaluates b^x. This page recovers x from b^x = a.
How does this relate to compound interest?
Solving A = P(1+r)^t for t is t = ln(A/P) / ln(1+r). Use the Compound Interest Calculator when the labels are dollars and a percent.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.