Logarithm Calculator
Work out log_b(x) for any base, the natural log ln(x), log₁₀ and log₂ at once, or switch to antilog mode to calculate bʸ. Every answer comes with the change-of-base working and a check.
Change-of-base steps
Steps
Examples
How to use the Logarithm Calculator
- Enter the number x you want the logarithm of. It must be positive. You can type decimals, fractions like 1/8, 2.5e-4 or e.
- Choose the base: e for ln, 10 for the common log, 2 for the binary log, or Custom to type any base b.
- Press Calculate (results also update as you type) to see log_b(x) along with ln, log₁₀ and log₂ of the same number.
- Read the steps to see the change-of-base formula ln(x) / ln(b) worked out.
- Switch to Antilog to go the other way: enter a base and a value y to get bʸ.
Logarithm rules
| Rule | Formula | Example |
|---|---|---|
| Definition | log_b(x) = y ⇔ bʸ = x | log₂ 32 = 5 because 2⁵ = 32 |
| Product | log_b(xy) = log_b x + log_b y | log 200 = log 2 + log 100 ≈ 0.30103 + 2 |
| Quotient | log_b(x/y) = log_b x − log_b y | ln(e³/e) = 3 − 1 = 2 |
| Power | log_b(xⁿ) = n · log_b x | log₂ 8⁴ = 4 × 3 = 12 |
| Change of base | log_b x = ln x / ln b | log₅ 100 = 4.60517 / 1.60944 ≈ 2.86135 |
| Log of 1 | log_b 1 = 0 | ln 1 = 0 |
| Log of the base | log_b b = 1 | log₁₀ 10 = 1 |
| Reciprocal base | log_b x = 1 / log_x b | log₈ 2 = 1/3 |
Worked example: log base 3 of 20
- Ask: 3 to what power gives 20? Since 3² = 9 and 3³ = 27, the answer is between 2 and 3.
- Change of base: log₃ 20 = ln 20 / ln 3.
- ln 20 ≈ 2.9957323 and ln 3 ≈ 1.0986123.
- 2.9957323 / 1.0986123 ≈ 2.7268330.
- Check: 3^2.7268330 ≈ 20.000000.
Common logarithm values
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 0.5 | −0.30103 | −0.69315 | −1 |
| 1 | 0 | 0 | 0 |
| 2 | 0.30103 | 0.69315 | 1 |
| e ≈ 2.71828 | 0.43429 | 1 | 1.44270 |
| 3 | 0.47712 | 1.09861 | 1.58496 |
| 5 | 0.69897 | 1.60944 | 2.32193 |
| 10 | 1 | 2.30259 | 3.32193 |
| 100 | 2 | 4.60517 | 6.64386 |
| 1000 | 3 | 6.90776 | 9.96578 |
Frequently asked questions
What is a logarithm?
A logarithm is the power you must raise the base to in order to get the number. log₂ 8 = 3 because 2³ = 8, and log₁₀ 1000 = 3 because 10³ = 1000.
How do I calculate a log with a different base?
Use the change-of-base formula: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). For example log₃ 20 = ln 20 / ln 3 = 2.995732 / 1.098612 ≈ 2.726833.
What is the difference between log and ln?
ln is the natural logarithm with base e ≈ 2.71828. log usually means base 10 in science and on calculators, but in computer science it often means base 2. This calculator shows all three for the same number.
Why can't I take the log of 0 or a negative number?
A positive base raised to any real power is always positive, so no real exponent gives 0 or a negative number. log(0) tends to minus infinity and logs of negative numbers are only defined with complex numbers.
What is an antilog?
The antilog is the inverse of a logarithm: antilog_b(y) = bʸ. For base 10, antilog(2.5) = 10^2.5 ≈ 316.2278. For the natural log, the antilog is eʸ.
Why can't the base be 1?
1 raised to any power is 1, so log base 1 could never give any other number. The base must be positive and different from 1.
What is log base 2 used for?
log₂ tells you how many times you can halve a number or how many bits are needed. log₂ 1024 = 10, so 10 bits can store 1024 different values.