Log Calculator - Base 10, e, 2, and Custom Logs
Calculate common, natural, binary, and custom-base logarithms instantly for any valid positive number.
Choose a log type, enter the value x, and get the logarithm with the correct notation right away.
Log Calculator - Base 10, e, 2, and Custom Logs
Calculate common, natural, binary, and custom-base logarithms instantly for any valid positive number.
About the log calculator
A logarithm answers the question, "What exponent gives this number?" If 10³ = 1000, then log₁₀(1000) = 3. That simple relationship is why logarithms are so useful in algebra, finance, statistics, computer science, chemistry, signal processing, and any subject where numbers grow or shrink multiplicatively instead of linearly. Rather than working directly with huge powers, you can convert multiplication into addition, powers into products, and exponential growth into straight-line thinking.
This log calculator covers the four log types most people need in practice. Common log, or base 10, is common in scientific notation and pH-style reasoning. Natural log, written ln, uses the constant e and shows up in calculus, compound growth, continuous decay, probability, and differential equations. Binary log, or base 2, is especially helpful in computing because doubling and halving map cleanly to powers of two. The custom-base mode lets you evaluate log_b(x) for any valid base b, which is useful when coursework, algorithms, or engineering formulas use a base other than 10, e, or 2.
The domain rules matter. The input x must be greater than zero because no real logarithm exists for zero or negative numbers. In custom-base mode, the base must also be greater than zero and cannot equal 1. A base of 1 fails because 1 raised to any power is still 1, so there is no unique exponent to solve for. Internally, the calculator uses JavaScript's built-in logarithm functions for base 10, base e, and base 2, and it uses the change-of-base identity log_b(x) = ln(x) / ln(b) for any other valid base.
One practical benefit of a dedicated log calculator is speed with interpretation. You can test powers, check homework, compare scales, and verify formulas without doing repeated transformations by hand. For example, log₂(1024) tells you how many times you can divide by two to reach 1, while ln(e²) immediately collapses to 2 because the natural log and the exponential function are inverse operations. Once you see logs as exponents in disguise, the notation stops feeling abstract.
Whether you are solving equations like 3^x = 81, reading a graph on a logarithmic scale, estimating doubling times, or converting a formula from one base to another, this log calculator gives you a fast and reliable way to compute the value and focus on what the result means in context.
Log calculator examples
These examples show the most common logarithm types and the kind of answers you can expect.
| Input | Result | Explanation |
|---|---|---|
| log₁₀(1000) | 3 | Because 10³ = 1000, the common logarithm of 1000 is 3. |
| ln(e²) | 2 | Natural log reverses powers of e, so ln(e²) returns the exponent 2. |
| log₂(64) | 6 | Since 2⁶ = 64, the binary logarithm equals 6. |
| log₃(81) | 4 | In custom-base mode, 3⁴ = 81, so the log is 4. |
How to use the log calculator
- Enter the positive value x in the number field.
- Choose whether you want a common log, natural log, binary log, or a custom-base logarithm.
- If you choose Custom base, enter the base value b and make sure it is positive and not equal to 1.
- Click Calculate to see the logarithm written in the correct notation.
- Use Reset to clear the fields and try another base or another number.
Log calculator FAQ
What does a logarithm mean?
A logarithm is the exponent you must raise a base to in order to get a target number. For example, log₁₀(100) = 2 because 10² = 100.
Why must x be greater than zero?
Real-valued logarithms are defined only for positive inputs. Zero and negative values do not produce real-number log results.
Why can’t the custom base be 1?
If the base is 1, every power is still 1, so there is no unique exponent that can produce different values of x. A base of 1 would make the logarithm undefined for any input other than 1.
When should I use ln instead of log₁₀?
Use ln when formulas involve e, continuous growth, decay, or calculus. Use log₁₀ when you want base-10 scaling or scientific notation contexts.
How is a custom-base log calculated?
The calculator uses the change-of-base formula log_b(x) = ln(x) / ln(b), which converts any valid base to natural logarithms under the hood. This means any base you enter is handled with the same level of precision as the built-in log functions.