THE NUMORIX GUIDE
How to use the Logarithm Calculator
Last reviewed September 14, 2026
What this calculator does
Compute log base b of a with log_b(a) = ln(a)/ln(b).
Formula and method
Compute log base b of a with log_b(a) = ln(a)/ln(b). The UI shortcuts use base 10 for log10 and Math.E for ln; a custom base must be positive and not 1.
Variables and inputs
Value defaults to 100. Mode defaults to log10; ln uses the natural logarithm, and Custom Base reveals a base field defaulting to 2.
Worked example
In log10 mode, log10(100) = 2 because 10^2 = 100. In custom base mode, log2(32) = ln(32)/ln(2) = 5 because 2^5 = 32.
How to interpret the result
A logarithm answers the question: what exponent on the base produces the value? It reverses exponentiation.
Common mistakes to avoid
Do not use a nonpositive value or base 1. Distinguish log10 from ln, and remember that a custom base is not the same as multiplying by that number.
Assumptions and limitations
The engine does not restrict custom bases to greater than one, so valid bases between zero and one are accepted but produce decreasing logarithms. Complex logarithms are not supported.
Practical use and checks
A logarithm answers the reverse-exponent question: what power of a base produces the value? Enter value 100 in base-10 mode as a check; the result should be 2 because 10 squared is 100. In custom-base mode, value 32 with base 2 should return 5 because 2 to the fifth power is 32. Use this relationship to check growth exponents, signal scales, or the number of repeated doublings needed to reach a target. The input value must be positive, and a custom base must be positive and not equal to 1. A base between 0 and 1 is mathematically allowed but reverses the direction of the function, so it should be used deliberately. Do not treat log10 and natural log as interchangeable, and do not read a log as a percentage without a model connecting it to one. The calculator uses the change-of-base formula with natural logarithms and supports real results only. A negative or zero value therefore has no ordinary output, while a base close to 1 can magnify rounding and measurement error. Check the result by raising the selected base to the displayed power, especially when deciding how many periods of exponential growth are represented.