How this instrument works
A logarithm answers a specific question: what power must a given number be raised to, to reach a target value? logₐ(x) = y means aʸ = x. Rather than needing a separate formula or button for every possible a, this calculator uses the change-of-base identity, logₐ(x) = ln(x) ⁄ ln(a), which computes any such logarithm using only the natural logarithm (built on e) — a single, general-purpose route to any answer at all.
The identity works because logarithms convert multiplication into addition in a very specific, provable way: taking the natural log of both sides of aʸ = x gives y·ln(a) = ln(x), and dividing through by ln(a) isolates y = ln(x) ⁄ ln(a) directly — exactly the formula used here, derived from the defining property of logarithms rather than assumed.
Two special cases get their own name and shorthand elsewhere: a = 10 (the 'common' logarithm, log₁₀, used throughout science for orders of magnitude like the Richter and pH scales) and a = e ≈ 2.71828 (the 'natural' logarithm, ln, which appears constantly in calculus and continuous growth models). This generic calculator handles both of those, and every other value, through the identical change-of-base approach.
- Enter the logarithm's base into the Base field.
- Enter the number you want the logarithm of into the x field.
- Read logₐ(x): the sheet applies the change-of-base identity directly.
Worked example — log base 2 of 8
log₂(8) computes to exactly 3, since 2³ = 8 — the change-of-base formula, ln(8) ⁄ ln(2), lands on exactly 3 despite neither ln(8) nor ln(2) being a clean number on its own, since their ratio is exact.
log₁₀(1000) computes to exactly 3, since 10³ = 1000 — the common logarithm, used throughout science for orders-of-magnitude comparisons. And log₅(1) computes to exactly 0 no matter which valid number sits underneath, since anything nonzero raised to the power 0 always equals 1.
Questions
What does a logarithm mean?
logₐ(x) answers 'what power must a be raised to, to get x?' It's the inverse operation of exponentiation — if aʸ = x, then logₐ(x) = y, by definition.
What is the change-of-base formula?
logₐ(x) = ln(x) ⁄ ln(a) — it converts a logarithm using any value of a into an equivalent calculation using only the natural logarithm, so a single formula handles every possibility rather than needing one per case.
Why can't a be equal to 1?
Because ln(1) = 0, and dividing by 0 in the change-of-base formula is undefined — this also makes sense directly, since 1 raised to any power always equals 1, so 'what power of 1 gives x' has no answer for any x other than 1 itself.
What is the logarithm of a number equal to a itself?
Always exactly 1 — logₐ(a) = 1 no matter which valid value a takes, since a¹ = a by definition. This is a useful quick sanity check whenever verifying a logarithm calculation by hand.
How is this different from ln or log₁₀?
ln and log₁₀ are logarithms with that underlying value already fixed (e and 10 respectively). This calculator generalizes the idea to any value at all, using the change-of-base identity to reach the same answer those fixed versions would give when their specific figure is entered here.