Logarithm Calculator
Use this calculator for exponential models, compound growth and decay, pH and decibel scales, and algebra practice where logarithms isolate unknown exponents.
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Result
The formula
A logarithm answers the question: “To what power must base b be raised to get x?” The core identity is y = log_b(x), equivalent to x = b^y. You can solve any one of these variables when the other two are known.
Worked example
- Given x = 64 and b = 2.
- Apply y = log_b(x) = ln(64)/ln(2).
- Result: y = 6.
Result: log₂(64) = 6
How this logarithm calculator works
Each mode applies one algebraic form of the same exponential relationship.
Find logarithm y = log_b(x)
Use this mode when x (value) and b (base) are known. Domain rules: x > 0, b > 0, and b ≠ 1.
Find value x = b^y
Use this mode when base b and exponent y are known. The result gives the value generated by exponential growth or decay.
Find base b = x^(1/y)
Use this mode when value x and exponent y are known. Constraints: x > 0 and y ≠ 0. Also b must remain positive and not equal to 1.
Input validity and edge cases
Logarithms are undefined for non-positive x or invalid bases. If a combination is outside the valid domain, the calculator hides the result instead of showing a misleading value.
Interesting facts
Changing base does not change meaning
log_b(x) can be computed with natural logs: ln(x)/ln(b). This is why calculators can evaluate logs for any valid base.
ln(x) is just log base e
The natural logarithm uses e ≈ 2.71828 as the base and appears frequently in finance, biology, and differential equations.
Log scales compress huge ranges
pH, Richter, and decibel scales use logarithms so very large multiplicative differences can be compared on smaller numeric ranges.
Used in real-world calculations
Logarithms are used to compute pH in chemistry, decibel levels in acoustics, earthquake magnitude scales, and compound-growth timelines in finance.
Frequently asked questions
You must have x > 0, b > 0, and b ≠ 1. Otherwise the logarithm is undefined in real numbers.
Yes. A negative exponent means reciprocal growth, so x becomes a fraction when b > 1.
Because 1 raised to any power is always 1, so it cannot produce unique logarithm values.
ln means base e. In many contexts “log” means base 10, but in mathematics the base should be stated explicitly.
Yes. Use “Solve for base” mode with known x and y to compute b = x^(1/y).