This log calculator returns log_base(value) using change of base. Enter a positive value and a valid base. The default value 100 with base 10 returns 2 because 10² = 100.
Algebra, pH style chemistry homework, and computer science orders of magnitude all lean on logs. For converting large counts into coefficient form after a log discussion, see the scientific notation page when it is published, or use the percentage calculator for percent change problems.
How the formula works
log_b(a) = ln(a) / ln(b). The value must be positive. The base must be positive and not equal to 1.
Worked example
Value 100, base 10. ln(100)/ln(10) = 2.
| Input | Value |
|---|---|
| Value | 100 |
| Base | 10 |
| Logarithm | 2 |
How to use the fields
- Value is the number inside the log.
- Base is the logarithm base.
Meaning of the result
The logarithm is the exponent that raises the base to the value. Saying log10(100) = 2 is the same as saying 10² = 100.
Common mistakes
- Swapping value and base
- Using base 1
- Entering zero or a negative value
- Expecting log to return a percent
Other common bases
Base 10 is common in general science. Base 2 appears in computing. Base e appears in continuous growth. Change only the base field to switch stories without rewriting the value.
Classroom practice
Ask what power of 10 makes 1000. Students should say 3, then confirm with value 1000 and base 10.
Try base 2 with value 8 to get 3, reinforcing that logs return exponents.
Checking with powers
After you read a log result, raise the base to that result and confirm you recover the value. For the default, 10² returns 100.
Equations with logs
Solving b^x = a is the same as x = log_b(a). Use the calculator to evaluate the log once the equation is in that form.
For equations with log already present, isolate the log expression first, then evaluate.
Growth and scales
Logs turn multiplicative scales into additive ones. Earthquake magnitudes and some audio measures use log ideas for that reason. This tool evaluates the bare logarithm so you can plug it into those larger formulas.
Always state the base when you report a log in mixed company, because log without a base is ambiguous across fields.
Calculator precision
Change of base uses floating point ln values. For clean textbook cases like log10(100), you should see an exact integer. Messy inputs may show long decimals that you should round to the course standard.
If a result looks slightly off from an integer, check whether the value and base truly match a perfect power.
Graph intuition
The log function grows quickly at first for bases greater than 1, then more slowly. That shape explains why log scales compress large ranges on charts.
Plotting is outside this calculator, but knowing the shape helps you trust that log10(100) = 2 and log10(1000) = 3 are evenly spaced in log space while the raw values are not.
For 0 less than value less than 1, real logs are negative. Try value 0.01 with base 10 to see -2.
Identity checks
Useful identities include log_b(xy) = log_b(x) + log_b(y) and log_b(x^k) = k log_b(x) for positive x. Evaluate pieces with this calculator to verify homework algebra.
Change of base also implies log_b(a) = 1 / log_a(b). Try base and value swapped on clean examples to see the reciprocal relationship.
Base e quick path
If you need natural log, set the base near 2.718281828 or whatever e approximation your course allows. The change of base path still applies.
Many science formulas already assume ln. Confirm whether your homework wants log10 or ln before you enter the base.
Limitations
Real logs only. No complex branch, no log of negative inputs, and no automatic equation solving beyond a single evaluation.