Logarithm Calculator.
Calculate common log, natural log, binary log and custom base logarithms with formulas, chart and examples.
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Formula
A logarithm answers this question: base raised to what power gives the value?
Logarithm Chart
Common Log Values
| Expression | Answer | Exponential Form |
|---|---|---|
| logâ‚â‚€(10) | 1 | 10¹ = 10 |
| logâ‚â‚€(100) | 2 | 10² = 100 |
| log₂(8) | 3 | 2³ = 8 |
| logâ‚‚(32) | 5 | 2âµ = 32 |
| ln(e) | 1 | e¹ = e |
| ln(1) | 0 | eâ° = 1 |
Logarithm Guide
Logarithms are the inverse of exponents. If 10² = 100, then log10(100) = 2. This means the logarithm tells us the power needed to produce a number from a given base.
Logarithms are useful whenever numbers grow or shrink quickly. They appear in compound growth, sound intensity, earthquake scales, pH levels, computer science, algorithms and calculus.
The most common logarithms are common log, natural log and binary log. Common log uses base 10, natural log uses base e, and binary log uses base 2.
Log Base Comparison
| Type | Notation | Used In |
|---|---|---|
| Common Log | logâ‚â‚€(x) | Science, pH, decibels, base-10 scales |
| Natural Log | ln(x) | Calculus, growth, decay, finance |
| Binary Log | logâ‚‚(x) | Computer science, bits, algorithms |
| Custom Base | logᵦ(x) | Any base conversion or exponential model |
Examples
Worked Example
logâ‚â‚€(1000)
Ask: 10 raised to what power equals 1000?
Worked Example
logâ‚‚(64)
Ask: 2 raised to what power equals 64?
Worked Example
ln(e²)
Natural log uses base e.
FAQs
1. What is a logarithm?
A logarithm tells you what power a base must be raised to in order to get a number.
2. What is log base 10?
Log base 10, written as log10(x), uses 10 as the base. For example, log10(100) = 2.
3. What is natural log?
Natural log, written as ln(x), uses Euler’s number e as the base.
4. What is binary log?
Binary log uses base 2 and is common in computer science.
5. What is the change of base formula?
log?(x) = ln(x) ÷ ln(b). This lets you calculate logarithms of any base.
6. Can logarithm input be negative?
No. In real numbers, the log input must be greater than zero.
7. Can log base be 1?
No. A logarithm base cannot be 1 because powers of 1 never change.
8. What is log10(1000)?
log10(1000) = 3 because 10³ = 1000.
9. What is ln(e)?
ln(e) = 1 because e¹ = e.
10. What is log2(32)?
log2(32) = 5 because 25 = 32.
11. Why are logarithms useful?
Logarithms help solve exponential equations and are used in science, finance, computing and engineering.
12. Does this calculator show charts?
Yes. The page includes a logarithm chart for log10(x), ln(x) and log2(x).
13. Is log the opposite of exponent?
Yes. Logarithms and exponents are inverse operations.
14. What is log of 1?
log?(1) is always 0 for any valid base b because b° = 1.
15. What is log of base itself?
log?(b) is always 1 because b¹ = b.
16. What is common log used for?
Common log is used in scientific notation, pH, decibels and base-10 scales.
17. What is natural log used for?
Natural log is used in calculus, growth models, decay models and continuous compounding.
18. Can this solve any base?
Yes. Enter any valid positive base except 1.
19. Is this calculator free?
Yes. HQCalc logarithm calculator is free.
20. Is data stored?
No. Calculations happen in your browser.