Calculator Inputs
Interactive Logarithm Graph
Compare logarithm and exponential inverse curves.
The vertical asymptote is x = 0. Move the pointer over the graph to inspect coordinates.
Calculation History
Your recent calculations appear here.
Formula Used
A logarithm answers an exponent question. The base indicates repeated multiplication. The argument must remain positive for real results.
How to Use
- Select the required calculation mode.
- Choose a standard base or enter a custom base.
- Provide the argument, exponent, equation constants, or batch values.
- Select the decimal precision and preferred number format.
- Press Calculate to view results and detailed steps.
- Copy, print, or export the completed calculation.
Worked Examples
| Expression | Result | Exponential form |
|---|---|---|
| log₁₀(1000) | 3 | 10³ = 1000 |
| ln(e⁴) | 4 | e⁴ = e⁴ |
| log₂(32) | 5 | 2⁵ = 32 |
| log₅(125) | 3 | 5³ = 125 |
| log₃(81) | 4 | 3⁴ = 81 |
| log₁₆(256) | 2 | 16² = 256 |
Logarithm Rules Reference
| Rule | Formula | Purpose |
|---|---|---|
| Product | log_b(xy) = log_b(x) + log_b(y) | Expands multiplication. |
| Quotient | log_b(x/y) = log_b(x) − log_b(y) | Expands division. |
| Power | log_b(xⁿ) = n·log_b(x) | Moves powers outside. |
| Root | log_b(x^(1/n)) = log_b(x)/n | Moves roots outside. |
| Change of base | log_b(x) = ln(x)/ln(b) | Evaluates any valid base. |
| Inverse | b^(log_b(x)) = x | Connects logarithms and exponents. |
Common Bases
| Base | Name | Common use |
|---|---|---|
| 10 | Common logarithm | Science, engineering, scales, and general calculations. |
| e | Natural logarithm | Growth, decay, calculus, finance, and probability. |
| 2 | Binary logarithm | Computing, algorithms, information theory, and data sizes. |
| 16 | Hexadecimal logarithm | Digital systems and hexadecimal scale comparisons. |
Practical Applications
Logarithms simplify quantities spanning large ranges. They appear in pH, decibels, earthquakes, and signal measurements. They also support growth and decay models.
Computer science uses binary logarithms extensively. Algorithm complexity often depends on repeated halving. Information theory also measures bits logarithmically.
Finance uses natural logarithms for continuous growth. Population models use related exponential relationships. Scientific analysis relies on these transformations.
Frequently Asked Questions
What is a logarithm?
A logarithm identifies the exponent needed to produce a number. It reverses exponentiation. The selected base controls the repeated multiplication.
What is the difference between log and ln?
Log usually means base ten in general calculators. Ln always means base e. Both follow the same logarithm rules.
Can the argument be zero?
No real logarithm exists for zero. Logarithm curves approach negative infinity near zero. The argument must remain strictly positive.
Can a logarithm base equal one?
No. One raised to any power remains one. Therefore it cannot create a useful inverse function.
Can the base be negative?
Negative bases are excluded for general real logarithms. Some isolated values may exist. Complex analysis requires additional definitions.
How does change of base work?
Divide one familiar logarithm by another. Use ln(x) divided by ln(b). This evaluates any valid positive base.
What is an antilogarithm?
An antilogarithm reverses a logarithm. It raises the base to the logarithm result. The output reconstructs the original argument.
Why can rounding affect verification?
Rounded logarithms lose small decimal details. Raising the base may show a tiny difference. Increasing precision reduces that difference.
Does the calculator support batch values?
Yes. Enter values separated by commas or line breaks. Each valid positive number receives a separate result.
Limitations and Notes
This calculator focuses on real-number logarithms. Complex logarithms require branch selection and specialised notation. Extremely large powers may exceed PHP numeric limits.
Equation templates cover common logarithmic forms. They do not parse every symbolic expression. Use the displayed steps to verify domain restrictions.