How this instrument works
Expanding a logarithm means splitting a single log of a product into a sum of two simpler logs: ln(a × b) rewrites as ln(a) + ln(b), one term for each factor. That direction matters — this sheet takes a product buried inside one logarithm and pulls it apart, the opposite move from folding several log terms back into one. The two sides are never approximately equal; they are the same number written two different ways, which is exactly what this calculator lets you confirm for any a and b you choose.
The identity has a tidy proof once you picture ln(x) as the area under the curve 1/t, measured from t = 1 out to t = x. The area out to a × b splits naturally at t = a into two pieces: the first piece is exactly ln(a). Stretch the second piece by dividing every t-value by a — the curve 1/t is built precisely so that this rescaling leaves the area unchanged — and what remains is the area from 1 to b, which is ln(b). Cut a region in two and rescale one part; the areas add, and so do the logs.
The rule only holds where both logs are defined, so a and b must each be positive; a product of a negative and a positive number has no real logarithm to expand, even though the product itself might be negative or positive. Near a = 1, ln(a) sits close to zero, so the expanded sum sits close to ln(b) alone — a quick sanity check when one factor barely changes the product at all.
- Enter the first factor into a, in log(a × b) — any positive number, whole or decimal.
- Enter the second factor into b, in log(a × b); together a and b form the product sitting inside the original logarithm.
- Read ln(a) + ln(b) for the expanded result: two separate natural logs, computed independently and added.
- Check it against ln(a × b) (for comparison), which multiplies a and b first and takes one logarithm — the two figures should match to the last displayed digit.
Worked example — 4 times 5 expands to ln(20)
Set a to 4 and b to 5. Expanded returns ln(4) + ln(5) = 1.3862943611198906 + 1.6094379124341003 = 2.995732273553991 — two logarithms computed separately and added together by hand, or by this sheet.
Combined multiplies first: 4 × 5 = 20, then takes one logarithm, ln(20) = 2.995732273553991. The two results land on the identical fifteen-decimal figure, because expanding never changes the value, only how many logarithms carry it — one term split into two, or two terms merged into one, the number underneath stays fixed.
Questions
What is the product rule for logarithms?
It states that ln(a × b) = ln(a) + ln(b) for any positive a and b: the log of a product equals the sum of the individual logs. This calculator's Expanded and Combined fields are that rule's left and right sides, computed independently so you can see them land on the same figure.
Why can't a or b be zero or negative here?
No real power of e ever produces zero or a negative number, so a natural logarithm simply has nothing to return once its argument drops to zero or below. If either factor were zero or negative, ln(a) or ln(b) alone would already break, well before the product rule ever gets a chance to apply.
Where does ln(a × b) = ln(a) + ln(b) actually come from?
Picture ln(x) as the area under the curve 1/t from t = 1 to t = x. The area out to a × b splits at t = a into a piece equal to ln(a), plus a second piece that — after rescaling every t by a, which the curve 1/t leaves unchanged in area — equals ln(b). Two regions add to one; two logs add to one.
Does this rule change if I use log base 10 instead of natural log?
No — log(a × b) = log(a) + log(b) holds for every base, common or natural, as long as both terms share the same base throughout. This calculator uses natural logs specifically, but swapping in base 10 or base 2 changes only the numbers involved, never whether the rule applies.
What mistake do people most often make when expanding a logarithm?
Applying the product rule to a sum instead of a product — writing ln(a + b) as ln(a) + ln(b), which is false in general. The rule only ever splits a logarithm of a MULTIPLIED pair of terms; a logarithm of a sum has no simpler equivalent form at all.
How is expanding a logarithm different from condensing one?
Expanding starts from one logarithm of a product and splits it into separate terms, as this sheet does with ln(a × b). Condensing runs the opposite direction, folding two or more separate log terms back into a single logarithm — the same identity, read from right to left instead of left to right.