Log Calculator

Use this log calculator to find the exponent y in b^y = x or solve for x from a known base and log value.

What to solve

Pick what you want to find. Logs answer this question: what exponent makes the base turn into the number?

Base option

Choose the base b. Common log uses base 10. Natural log uses base e. Binary log uses base 2. Use Custom to type your own base.

Number x

This is the positive number inside the log. For real-number logs, x must be greater than 0.

Custom base b

Only used when Base option is Custom. For real-number logs, b must be greater than 0 and cannot equal 1.

Log value y

This is the exponent in b^y = x. Use this when you are solving for the number x.

Decimal places

Rounds the displayed answer. The calculator still computes using full precision first.

Result

Equivalent equation

Base used

Domain check

Did we solve your problem today?

For an expression such as log(x - 1), or a logarithmic equation with a linear expression, use the Logarithmic Expression and Equation Calculator.

How to use our Log Calculator

Choose What to solve: use Log value y from base b and number x if you want the logarithm, or Number x from base b and log value y if you want the original number.
Pick a Base option: common log is base 10, natural log ln is base e, binary log is base 2, or choose Custom base.
If you picked Custom base, enter Custom base b. For real-number logs, the base must be greater than 0 and cannot equal 1.
If you are solving for the log, enter Number x. This value must be greater than 0.
If you are solving for the number, enter Log value y. This is the exponent in b^y = x.
Choose Decimal places to control rounding in the displayed answer.
Click Calculate to see Result, Equivalent equation, Base used, and Domain check.
Sanity-check the answer: if Result is 2 with base 10, the Equivalent equation should read 10^2 = 100; if your number is between 0 and 1 with base greater than 1, a negative log result can still be correct.

Definitions

What to solve: The mode that decides whether you want the logarithm value y or the original number x.

Base option: The base b used in the logarithm. Common log uses 10, natural log uses e, and binary log uses 2.

Number x: The positive number inside the logarithm. In real-number logs, x must be greater than 0.

Custom base b: Your own base value. For real-number logs, b must be greater than 0 and cannot equal 1.

Log value y: The exponent in the statement b^y = x. This is what a logarithm gives you.

Result: The calculated answer for the active mode, rounded to the selected number of decimal places.

Equivalent equation: The same relationship written in exponent form so you can check that b^y = x.

Domain check: A quick message telling you whether your entries are valid for real-number logarithms.

Common mistakes and quick fixes

Mistake: Entering 0 or a negative value in Number x when solving for the logarithm.
Fix: For real-number logs, Number x must be greater than 0. Enter a positive number and recalculate.

Mistake: Setting Custom base b to 1.
Fix: Change Custom base b to any positive value other than 1, because 1^y is always 1 and does not make a valid real log base.

Mistake: Typing a value into Custom base b but leaving Base option on common log, natural log, or binary log.
Fix: If you want your own base, switch Base option to Custom base . Otherwise the calculator uses the preset base shown in Base used .

Mistake: Reading Result as the final number x when the mode is Log value y from base b and number x .
Fix: In that mode, Result is the exponent y. Check Equivalent equation to see how the exponent matches the number.

Mistake: Thinking a negative Result means the entry is invalid.
Fix: A negative log value is valid when Number x is between 0 and 1 and the base is greater than 1. Use Domain check to confirm the inputs are valid.

Mistake: Rounding too early and then comparing homework answers digit by digit.
Fix: Increase Decimal places before comparing results. The displayed Result is rounded at the end.

Limitations & Key Assumptions / Boundary Conditions

This calculator is limited to real-number logarithms. It does not calculate complex logs for negative values of Number x or invalid bases.
When solving for a log value, Number x must be greater than 0.
When using Custom base b, the base must be greater than 0 and cannot equal 1.
Preset bases 10, e, and 2 override any hidden custom-base entry, so hidden inputs do not affect the result.
Very large positive or negative values of Log value y can overflow or underflow normal floating-point math when solving for x.
Displayed answers are rounded by Decimal places, so the shown value may differ slightly from an unrounded classroom or software result.
Very small or very large answers may need scientific notation for correctness, even if that is harder to read.

Methodology

Core idea

A logarithm tells you the exponent. If y = log_b(x), then y is the power you put on base b to get x [1].

y = log_b(x) <=> b^y = x

How the calculator computes the log

For base 10, base e, base 2, or a custom base, the calculator finds the logarithm value with the change-of-base rule [2].

log_b(x) = ln(x) / ln(b)

Here, ln means natural logarithm, which uses base e. The value of e is about 2.718281828459045 [3].

How the calculator solves for the number

If you already know the exponent y and the base b, the original number is found with exponential form.

x = b^y

Worked mini-example

If What to solve is log value, Base option is common log, and Number x is 100, then the calculator solves for y in 10^y = 100.

y = log_10(100) = 2

10^2 = 100

If What to solve is number x, Custom base b is 5, and Log value y is 3, then:

x = 5^3 = 125

Domain checks used

For real-number logarithms, the inside number must be positive and the base must be positive but not 1 [1]. That is why the calculator blocks x less than or equal to 0, base less than or equal to 0, and base equal to 1.

x > 0

b > 0

b != 1

Assumptions behind the result

The calculator uses standard real-number logarithm rules and ordinary floating-point arithmetic. That means exact math ideas are being shown with rounded decimal output, and extreme inputs can produce overflow, underflow, or scientific notation in the displayed answer.