Logarithm Calculator

A logarithm answers one question: what power do I raise this base to, to get this number? This calculator finds it in any base, shows the common, natural and base-2 logs alongside, and works backward with the antilog.

How to Use It

1. Choose Logarithm or Antilog (Power).
2. Enter the Base: 10, 2, or e for the natural log.
3. For a logarithm, enter the Number. For an antilog, enter the Exponent.
4. Press Calculate. A logarithm also shows the same number’s log in bases 10, e and 2.

How It Works

A logarithm undoes a power:

Calculators only have buttons for base 10 and base e, so any other base uses the change-of-base rule:

The antilog goes the other way: it raises the base to the power, $b^y$.

Worked Example

Find $\log_2 8$. Since $2^3 = 8$, the answer is exactly 3. By change of base:

The antilog reverses it: $2^3 = 8$.

Tips and Common Mistakes

• Only positive numbers have real logs. $\log 0$ and the log of a negative number don’t exist.
• Check which log your class means. “log” is usually base 10, but some courses and programming languages use it for base e.
• Logs turn multiplying into adding. $\log(ab) = \log a + \log b$, which is why logs simplify big calculations.
• The base can’t be 1. Every power of 1 is 1, so it can’t reach any other number.
• Numbers below 1 have negative logs. $\log_{10} 0.01 = -2$ because $10^{-2} = 0.01$.