Quick answer: log₂(1,024) = 10 because 2^10 = 1,024; by change of base, ln(1,024) ÷ ln(2) = 6.93147181 ÷ 0.69314718 = 10.
How it's calculated
The logarithm answers "what power of b gives x?": log_b(x) = y means b^y = x. So log₁₀(1,000) = 3 and log₂(8) = 3.
Any base can be worked out from natural logs with the change-of-base formula: log_b(x) = ln(x) ÷ ln(b). The base must be positive and not 1, and x must be greater than 0.
log₁₀ (common log) is used for decibels, pH and orders of magnitude, ln (base e ≈ 2.71828) for continuous growth, and log₂ for bits and doubling. To go the other way, raise the base to the answer with the exponent calculator.
Worked examples
- Base 2: log₂(1,024) = 10
- Base 10: log₁₀(1,000) = 3
- Base 5: log₅(125) = ln(125) ÷ ln(5) = 3
Logarithm rules
| Rule | Example |
|---|---|
| log(xy) = log x + log y | log₁₀(20) = log₁₀ 2 + log₁₀ 10 = 1.30103 |
| log(x ÷ y) = log x − log y | log₂(32 ÷ 4) = 5 − 2 = 3 |
| log(xⁿ) = n·log x | ln(e³) = 3 |
| log_b(b) = 1, log_b(1) = 0 | log₇(7) = 1 |
Frequently asked questions
Why can't I take the log of 0 or a negative number?
No power of a positive base gives 0 or a negative number, so the logarithm is undefined there (in real numbers).
What is the difference between log and ln?
log usually means base 10 and ln means base e. In some maths and programming texts log means ln, so check the base.
How do I find an antilog?
The antilog of y in base b is b^y. For example the antilog of 2 in base 10 is 10² = 100.
What is log base 2 of 1024?
10, because 2 multiplied by itself 10 times is 1,024.
What is log₁₀ of 1,000,000?
6. With powers of ten, the common log just counts the zeros.
How are logs used for pH?
pH is minus the log of the hydrogen ion concentration, so every step on the scale is ten times stronger or weaker. Lemon juice at pH 2 is ten times more acidic than something at pH 3.