Logarithm Calculator

A logarithm answers the question: what power of the base gives this number? For example, log base 10 of 1000 is 3, because 10³ = 1000. This logarithm calculator works in any base, including the natural log with base e, and shows the change-of-base method for answers that are not whole numbers.

Type e for the natural log (ln).
Try:

How to use the logarithm calculator

  1. Enter the number x (it must be positive).
  2. Enter the base, such as 10 or 2, or type e for the natural log.
  3. Press Calculate to see the value and the steps.

Formula

log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x
log⁡bx=ln⁡xln⁡b\log_b x = \frac{\ln x}{\ln b}

Worked example: log₁₀ 1000

  • Number (x): 1000
  • Base: 10

✓ Answer checked

33
Step-by-step working (2 steps)

  1. Ask the question Definition of a logarithm

    The logarithm asks: what power of 10 gives 1000?

    log⁡10(1000)=y  ⟺  10y=1000\log_{10}(1000) = y \iff 10^{y} = 1000

  2. Find the power

    10 to the power 3 equals 1000.

    103=100010^{3} = 1000
Formulas used
log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = x
log⁡bx=ln⁡xln⁡b\log_b x = \frac{\ln x}{\ln b}
How this was checked
  • ✓ 10 raised to the answer ≈ 1000, the original number.

Frequently asked questions

What is a logarithm?

The power you raise the base to in order to get a number. log₂ 64 = 6 because 2⁶ = 64.

What is ln?

ln is the natural logarithm, which uses the base e ≈ 2.71828.

Why can't I take the log of a negative number?

A positive base raised to any real power is always positive, so no real power gives a negative number or zero.

Related calculators

Exponents & Powers Calculator

Raise any number to a power, including negative and fractional exponents.

Square Root Calculator

Find the square root of any number, simplified exactly (like √50 = 5√2) with the decimal value.

Related guides