About Logarithm Calculator
This logarithm calculator is free, needs no sign-up and runs in your browser, so the numbers you type are never uploaded. It computes the logarithm of any positive number to any positive base other than 1, plus the natural log ln, the common log log10, the binary log log2, the antilogarithm, and solves exponential equations of the form b^x = y.
When the number is an exact rational power of the base, the answer is exact: log base 8 of 4 is 2/3, and log base 2 of 1/8 is -3. Otherwise the answer is a decimal to 12 significant digits, with the change-of-base calculation shown using both natural and common logs so you can repeat it on any calculator.
Below the answer you get the same number in the three standard bases and a reference list of the logarithm rules. Domain problems are explained instead of showing a bare error: the log of zero or a negative number is not a real number, and the base cannot be 1.
How to use Logarithm Calculator
- Choose a mode: log of any base, ln, log10, log2, antilog or solve b^x = y.
- Enter the base (not needed for ln, log10 and log2) and the number. You can type fractions like 1/8, powers like 2^10 or the constant e.
- Read the exact or decimal answer at the top.
- Open the steps for the definition, the change of base and, when the number is an integer, the product and power rules applied to its prime factors.
- For an exponential equation such as 1.05^x = 2, choose Solve b^x = y.
Features
- Log of any base, with exact rational answers when the number is an exact power of the base.
- Natural log, common log and binary log as one-click modes.
- Antilogarithm: b raised to the power y, exact for integer exponents.
- Solve b^x = y, the form of compound interest, growth and half-life questions.
- Change-of-base steps using both ln and log10.
- Product and power rules shown on the prime factors of an integer argument.
- Related logs in bases e, 10 and 2, and a list of the logarithm rules.
- Plain explanations of domain errors.
The logarithm rules and change of base
A logarithm is the exponent that a base must be raised to in order to give a number: log_b(x) = y means b^y = x. Its main rules turn products into sums, quotients into differences and powers into multipliers.
A calculator usually has only ln and log10, so any other base is found with the change-of-base formula.
log_b(x) = ln(x) / ln(b) = log10(x) / log10(b)
log_b(xy) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) - log_b(y)
log_b(x^n) = n · log_b(x)
log_b(b) = 1, log_b(1) = 0Worked examples
log base 2 of 8 gives 3
2³ = 8, so the exponent is exactly 3.
log base 8 of 4 gives 2/3
8^(2/3) = (∛8)² = 4, so the answer is the exact fraction 2/3.
log base 2 of 40 gives ≈ 5.32192809489
Not an exact power. Change of base: ln 40 / ln 2. Also log₂40 = 3 + log₂5, since 40 = 2³ × 5.
ln 10 gives ≈ 2.30258509299
The natural log has base e, about 2.71828.
antilog base 10 of 2.5 gives ≈ 316.227766017
10^2.5 = 100 × √10, about 316.23.
Logarithm names
| Name | Base | Written | Often used in |
|---|---|---|---|
| Common logarithm | 10 | log x or log10 x | Decibels, pH, the Richter scale |
| Natural logarithm | e ≈ 2.71828 | ln x | Calculus, growth and decay |
| Binary logarithm | 2 | log2 x | Computing, information, bits |
| Any base | b | log_b x | General algebra |
Good to know
Precision and domain
Decimals use double-precision arithmetic and are shown to 12 significant digits. Exact answers are detected only for rational inputs and rational exponents with a denominator up to 12.
In many programming languages and some textbooks, "log" means the natural log. Here log means base 10 and ln means base e, the usual convention in school mathematics.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The numbers you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- To check a log, raise the base to your answer. It should give back the number.
- log of a number between 0 and 1 is negative, and log of 1 is 0 in every base.
- For doubling time at a rate r per period, solve (1 + r)^x = 2 with the Solve mode.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is a logarithm?
The logarithm of x to base b is the power you raise b to in order to get x. Since 2³ = 8, log base 2 of 8 is 3.
How do I calculate a log with a base my calculator does not have?
Use change of base: log_b(x) = ln(x) / ln(b), or the same with log10. The calculator shows both versions with your numbers filled in.
What is an antilog?
The antilogarithm undoes the log: antilog_b(y) = b^y. The antilog of 3 in base 10 is 1000.
What is the difference between ln and log?
ln is the natural logarithm with base e. log usually means base 10 in school mathematics, and some programming languages use log for the natural log.
Why can I not take the log of zero or a negative number?
No real power of a positive base gives zero or a negative number, so the real logarithm is undefined there. The calculator explains this instead of returning a number.
How do I solve an equation like 5^x = 40?
Take the log of both sides and divide: x = log(40) / log(5), about 2.292. Choose Solve b^x = y and enter 5 and 40.
Why can the base not be 1?
Every power of 1 is 1, so no exponent could give any other number. The base must be positive and not 1.
Popular Logarithm Calculator pages
- Natural Log CalculatorEnter a positive number to get ln(x), the logarithm to base e.
- Log Base 2 CalculatorEnter a positive number to get log2(x), how many times you can halve it.
- Log Base 10 CalculatorEnter a positive number to get log10(x), the common logarithm.
- Antilog CalculatorEnter a base and an exponent to get the antilogarithm, b raised to y.
- Change of Base CalculatorEnter a base and a number to see log_b(x) worked out with the change-of-base formula.
- Solve Exponential EquationsEnter a base b and a target y to solve b^x = y for x.