Formula
log_base(value) = ln(value) ÷ ln(base). The value must be positive and the base must be positive and not equal to 1.
Logarithm Calculator
Calculate a logarithm in any positive base, with natural-log and common-log cross-checks shown beside the result.
What-if check
Change-of-base proof
Any valid logarithm base can be calculated by dividing natural logs, which makes the base rule visible on the printed page.
| Check | Value | Note |
|---|---|---|
| value | 1,000 | must be positive |
| base | 10 | positive and not 1 |
| ln(value) | 6.90775528 | change-of-base numerator |
| ln(base) | 2.30258509 | change-of-base denominator |
| current output | 3 | log base 10 = 3 |
Visual proof
Result path
ln(1,000) ÷ ln(10) = 3. Natural log 6.90775528; common log 3.
Worked example
For value 1,000 and base 10, log10(1,000) = 3 because 10 × 10 × 10 = 1,000.
Professional note
Master’s Tip: read a logarithm as the exponent needed to reach the value. That keeps logs connected to exponent rules instead of feeling like a separate trick.
Regional and unit assumptions
Standard or basis: change-of-base logarithm arithmetic using natural logs internally.
Assumptions and limitations
- The input value must be greater than zero.
- The base must be greater than zero and not equal to 1.
- Results use browser floating-point precision.
- This page calculates numeric logs, not log-scale charts or statistical transformations.
Methodology & Accuracy
How this calculator is checked
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Formula used
log_base(value) = ln(value) ÷ ln(base). The value must be positive and the base must be positive and not equal to 1.
Standard or basis
Standard or basis: change-of-base logarithm arithmetic using natural logs internally.
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master's Tip
Master’s Tip: read a logarithm as the exponent needed to reach the value. That keeps logs connected to exponent rules instead of feeling like a separate trick.
Related calculators
Questions
What is a logarithm?
A logarithm answers: what exponent do I need on this base to get the value?
Why can’t I log a negative number here?
This calculator is for real-number logarithms. Negative values require complex-number methods.
What is natural log?
Natural log uses base e, approximately 2.718281828.
Calculation note
Logarithms turn multiplication, division and powers into addition, subtraction and multiplication, which made them important long before electronic calculators.