Logarithm Calculator
Compute the logarithm of a positive number in any base — base 10, base e (natural log), base 2 or anything else — using the change-of-base formula.
Calculator
- Any base (10, e, 2, …)
- Change-of-base formula
- Domain validation
- Power-back check
Result & calculation receipt
- Method
- Change-of-base formula using natural logarithms
- Formula
log_b(x) = ln(x) / ln(b)- Substitution
- log_10(1000) = ln(1000) / ln(10) = 6.907755 / 2.302585 = 3
- Steps
- ln(1000) = 6.90775528
- ln(10) = 2.30258509
- Divide: 3
- Verification
- 10^3 = 1000 ≈ 1000.
- Decimal result
- 3
- Engine
- math-solvers v1.4.0
How this calculator works
The logarithm answers "what power gives this number?". The calculator uses the change-of-base formula log_b(x) = ln(x) ÷ ln(b), which works for any valid base. It checks the domain (the value must be positive, and the base must be positive and not 1) and verifies the answer by raising the base to the result to recover the original value.
Worked example
log₁₀(1000) asks what power of 10 gives 1000; the answer is 3, since 10³ = 1000. log₂(8) = 3, because 2³ = 8. The change-of-base step shows ln(8) ÷ ln(2), and the check raises 2 to the result to return 8.
Key insight
A logarithm is the inverse of an exponent: log_b turns multiplication into addition and powers into products. That is why logs underlie decibels, pH, the Richter scale and any measurement that spans many orders of magnitude.
Frequently asked questions
What is a logarithm?
The logarithm log_b(x) is the exponent to which the base b must be raised to produce x. So log₁₀(1000) = 3 because 10³ = 1000.
What is the change-of-base formula?
log_b(x) = ln(x) ÷ ln(b) (any common logarithm works in place of ln). It lets you compute a logarithm in any base from natural logs, which is what the calculator does.
What is the difference between log and ln?
"log" usually means base 10 and "ln" means base e ≈ 2.718, the natural logarithm. Both are just logarithms in specific bases; this tool lets you choose any base.
Why must the value be positive?
No power of a positive base gives zero or a negative number, so the logarithm is only defined for arguments greater than 0. Zero or negative inputs return a domain error.
Why can’t the base be 1?
Every power of 1 is 1, so log base 1 cannot single out a value — it is undefined. The base must be positive and different from 1.
Methodology & verification
Logarithms are computed with the change-of-base formula log_b(x) = ln(x)/ln(b). The domain is validated: the argument must be positive and the base positive and not equal to 1; other inputs return structured errors. The result is verified by raising the base to the computed exponent and confirming it recovers the argument, and the natural-log intermediates are shown.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.