Logarithm calculator

What a logarithm is

log_b(x) is the exponent you need to put on b to produce x. Since 10³ = 1000, log₁₀(1000) = 3. That is the entire definition — everything else follows from it.

Because it inverts exponentiation, a logarithm turns multiplication into addition. That property is why logarithms existed for three centuries before electronic calculators: multiplying two large numbers became adding two table lookups, which is the principle a slide rule runs on.

The three bases you will meet

• Base 10 — the common log, written log. Used wherever numbers span orders of magnitude: pH, decibels, the Richter scale.
• Base e — the natural log, written ln, where e ≈ 2.71828. It appears wherever something grows or decays at a rate proportional to its current size, which covers compound interest, radioactive decay and population growth.
• Base 2 — the binary log. Counts halvings and doublings, so it shows up throughout computer science: the depth of a binary search over n items is log₂(n).

Note that "log" without a base is ambiguous. Mathematicians and many programming languages mean base e; engineers and calculators usually mean base 10. In this calculator the base is always explicit.

Change of base

Most calculators only have log and ln buttons, but you can get any base from either:

log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)

So log₅(300) = ln(300) / ln(5) = 5.7038 / 1.6094 = 3.544. This calculator uses exactly that, which is why it can accept any base you type.

The rules worth memorising

• log(ab) = log a + log b
• log(a/b) = log a − log b
• log(aⁿ) = n log a
• log_b(b) = 1, and log_b(1) = 0 for any valid base

The third is the one that does the most work in practice, because it pulls an exponent down into a coefficient — which is how you solve for a variable stuck in an exponent. To find t in 2^t = 50: take logs of both sides to get t log 2 = log 50, so t = log 50 / log 2 = 5.64.

Why zero and negatives fail

There is no power of 10 that gives 0 or a negative number. As x approaches 0 the logarithm falls without limit towards −∞, and it never arrives. Base 1 fails for a related reason: 1 raised to anything is 1, so log₁(x) has no answer unless x is 1, and then it has infinitely many.

Common questions