Math Tools
Log Calculator
Calculate logarithms with a custom base, common logs, and natural logs.
Log Calculator
Calculate logarithms with a custom base, base 10, and base e.
Calculate log base b of a value using log_b(x).
Formula
log_b(x) = ln(x) / ln(b)
Result
2
Recommended Calculator
Need a physical scientific calculator for logs?
A scientific calculator with log, ln, exponent, and trigonometry keys can be useful for exams, homework, engineering, and science classes.
This section may contain affiliate links.
What this calculator does
This calculator finds logarithms using a custom base, base 10, or base e. It is useful for algebra, exponential equations, and science problems.
What is a log calculator?
A log calculator is a tool that helps you calculate logarithms quickly without solving them by hand. It can find a logarithm with a custom base, a common logarithm with base 10, or a natural logarithm written as ln.
This is useful in algebra, precalculus, statistics, chemistry, physics, finance, and computer science because logarithms appear in exponential growth, decay, pH values, sound intensity, compound interest, and many scientific formulas.
Logarithm formulas and notation
Custom base logarithm
log_b(x) = ln(x) / ln(b)
Common logarithm
log(x) = log₁₀(x)
Natural logarithm
ln(x)
Log type comparison table
| Type | Notation | Base | Common use |
|---|---|---|---|
| Custom base log | log_b(x) | Any valid base b | Algebra, exponential equations, base conversion |
| Common log | log(x) or log₁₀(x) | 10 | Scientific notation, engineering, powers of 10 |
| Natural log | ln(x) | e | Growth, decay, calculus, finance, science |
Worked logarithm examples
These example logarithm calculations show how common log, natural log, and custom base logs behave.
| Example | Result |
|---|---|
| log₂(8) | 3 |
| log₃(27) | 3 |
| log₅(125) | 3 |
| log₂(32) | 5 |
Quick log reference table
Use this small log table as a fast reference for common logarithm values.
| Expression | Value |
|---|---|
| log₂(4) | 2 |
| log₂(16) | 4 |
| log₃(9) | 2 |
| log₁₀(1000) | 3 |
Valid logarithm inputs and restrictions
Logarithms are only defined for certain values. Use this table to check whether your input and base are valid before calculating.
| Rule | Requirement | Example |
|---|---|---|
| Input value | x must be greater than 0 | log(100) is valid, log(0) is not valid |
| Base | Base must be greater than 0 | log₂(8) is valid, log₋₂(8) is not valid |
| Base cannot be 1 | Base must not equal 1 | log₁(10) is undefined |
How to calculate logarithms step by step
First, choose the logarithm type you want to calculate. Use custom base when you need log base b of x, common log when you need log base 10, and natural log when you need ln with base e.
Then enter the value. If you selected a custom base logarithm, enter the base as well. The calculator applies the correct logarithm formula automatically and returns the result instantly.
For a custom base, the calculator uses the change of base formula: ln(x) divided by ln(b). This makes it possible to calculate logs for almost any valid base.
Difference between log and ln
The term log usually means the common logarithm with base 10, especially in school math, engineering, and many calculator contexts. The term ln means the natural logarithm with base e.
Both are logarithms, but they use different bases. For example, log(100) equals 2 because 10² = 100, while ln(e²) equals 2 because e² = e². Knowing whether you need log or ln is important when using formulas in algebra, calculus, finance, and science.
Where a logarithm calculator is useful
An online log calculator is useful for solving exponential equations, simplifying algebra problems, checking homework, and working through scientific calculations. Logarithms are common in pH formulas, earthquake magnitude scales, sound intensity, compound growth models, and statistics.
It is also useful for students and professionals who want quick, accurate answers without doing manual change-of-base steps every time.
Log calculator FAQ and common logarithm questions
How do you calculate a logarithm with a custom base?
Use the change of base formula: log base b of x equals ln(x) divided by ln(b). This lets you calculate a logarithm with nearly any valid base.
What is the difference between log and ln?
Log usually means base 10, while ln means base e. They are both logarithms, but they are based on different numbers.
What is log base 10 used for?
Log base 10 is common in algebra, engineering, scientific notation, and problems involving powers of 10.
What is the natural logarithm?
The natural logarithm is written as ln(x) and uses base e, where e is approximately 2.71828.
Can a logarithm be negative?
Yes. A logarithm can be negative when the input value is between 0 and 1, as long as the base is valid and the value is positive.
Related math calculators
Explore related tools for arithmetic, algebra, geometry, statistics, and sequences.
Scientific Calculator
Solve advanced calculations with exponents, roots, logs, and trigonometry.
Exponent Calculator
Calculate powers and exponents for any number.
Root Calculator
Find square roots, cube roots, and nth roots.
Scientific Notation Calculator
Convert between standard form and scientific notation.
Matrix Calculator
Perform matrix addition, subtraction, multiplication, and more.