A logarithm asks for an exponent
The statement log_b(x) = y means b raised to y equals x. The base tells you which repeated scaling process is being reversed. For example, log base 2 of 32 is 5 because 2⁵ = 32.
On these calculators, log without another base means base 10. The notation ln means base e, where e is approximately 2.71828.
Use change of base consistently
For a custom base b, divide ln(x) by ln(b). You can also use common logs in both numerator and denominator. Mixing bases between the two parts does not give the same formula.
Real logarithms require x > 0, b > 0 and b ≠ 1. A base between zero and one is valid and reverses the usual increasing behavior.
Do not distribute across addition
Logarithms turn multiplication into addition: log_b(xy) = log_b(x) + log_b(y) for positive x and y. They do not turn log_b(x+y) into a sum of separate logs.
A negative logarithm result is allowed. For a base greater than one, a positive input smaller than one has a negative logarithm.
LET’S WORK IT THROUGH
Find log base 2 of 64.
- Ask which power of 2 produces 64.
- 2⁶ = 64.
- Alternatively, ln(64)/ln(2) = 6.
THE ANSWERThe logarithm is 6.
YOUR TURN
A small check for understanding.
What is log base 10 of 0.01?
Reveal the answer
−2, because 10⁻² = 0.01.
Now try your own numbers.
The logarithm calculator includes input guidance and the limits of its supported calculations. Use it alongside the method above.
Open Logarithm calculator ↗A note on precision: preserve exact fractions where possible and round numerical results at the end. Found an unclear step? Prepare a correction report.