How this instrument works
Condensing logarithms runs the product and quotient rules in reverse: two added terms combine into the log of a product, and a subtracted term becomes a division inside that same logarithm, so log(x) + log(y) − log(z) collapses to log(xy ⁄ z) in one move. The reason addition of logs turns into multiplication of arguments traces back to what a logarithm actually is: the log of a number is the exponent that recovers it from a fixed base, and exponents add exactly when the values behind them multiply — a^m × a^n = a^(m+n) — so x and y multiply precisely when their exponents add.
The identity holds without approximation for any positive x, y, and z, because a logarithm is only ever defined where its argument is positive; z also cannot be zero, since dividing by it would be meaningless before a logarithm even enters the picture. Condensing exists as a step on the way somewhere, not an end in itself — folding three log terms into one matters because a lone logarithm on each side of an equation can be removed in a single stroke by exponentiating both sides, since log is a one-to-one function.
There is a pleasing symmetry in reversing a nearly 400-year-old trick. John Napier built logarithms in 1614 specifically so that multiplying two large numbers could be done by adding their logs instead, turning hard multiplication into easy addition — the whole point of log tables and slide rules for the next three centuries. Condensing runs that machinery backwards: instead of turning a product into a sum for easier computation, it turns a sum of logs back into the single product or quotient it always secretly represented.
- Type the value inside the first logarithm into x, in log(x) — this is the term being added.
- Type the value inside the second logarithm into y, in log(y) — it also adds, so it multiplies against x.
- Type the value inside the subtracted logarithm into z, in log(z) — subtraction here means this term divides instead.
- Read Condensed argument for xy ⁄ z — the number that stands in for all three original terms once collapsed into one logarithm.
Worked example — three terms collapse to log(20)
Suppose an equation carries log(6) + log(10) − log(3) and needs to become one term before you can exponentiate both sides. Enter 6, 10, and 3 into x, y, and z, and Condensed argument returns 6 × 10 ⁄ 3 = 20 — the three terms collapse to the single log(20).
That collapse is what makes the equation solvable in one step: because a logarithm is one-to-one, if the left side equals log(w) for some unknown w, condensing it to log(20) forces w = 20 directly — no separate handling of three terms, no guessing, just one exponentiation to remove the last logarithm.
Questions
Why does subtracting logarithms turn into division instead of subtraction?
Because log(x) − log(y) = log(x ⁄ y), subtracting one exponent from another is the same operation as dividing the numbers those exponents produce. The second exponent is the power that turns the base into y, so negating it turns the base into 1⁄y — and adding that to the first exponent multiplies x by 1⁄y, which is x ⁄ y. That's exactly why z in this calculator divides into xy rather than being subtracted from it.
What's the most common mistake when condensing log(x) + log(y)?
Writing log(x + y) instead of the correct log(xy). Addition of the two terms corresponds to multiplication of their arguments, not addition of them — the sum of two log terms is never the log of their sum. The mix-up is understandable since the outer operation really is addition; it's only the arguments underneath that get multiplied together.
Does the base of the logarithm change how condensing works?
No — log(x) + log(y) − log(z) = log(xy ⁄ z) holds for any base, whether it is natural log, log base 10, or log base 2, as long as every term in the expression shares that same base. Mixing bases inside one expression breaks the rule, since it only ever combines logs already written in one common base.
What values are x, y, and z allowed to take?
All three must be positive, since a real logarithm is only defined for a positive argument — condensing three valid log terms always produces another valid one, xy ⁄ z, as long as z is not zero. If any input were zero or negative, the original expression was never a real number to begin with, condensed or not.
How is condensing different from expanding a logarithm?
Expanding runs this identity the other way: it takes one logarithm of a product or quotient, such as log(xy ⁄ z), and splits it into the separate terms log(x) + log(y) − log(z), which helps when differentiating a log or estimating it from known values. Condensing does the opposite, folding several terms into one before solving an equation.
Where does the rule for combining logarithms actually come from?
It comes from the exponent rules that define a logarithm in the first place: log(x) is the power a base is raised to in order to get x, and powers add when their underlying values multiply, a^m × a^n = a^(m+n). John Napier built the first log tables in 1614 for the reverse reason, to replace hard multiplication with easy addition, and condensing simply runs that centuries-old shortcut backwards.