Condense Logarithms Calculator
2 log x + 3 log y − log z is three separate logarithms. Pushed back together they become one: log(x²y³/z).
That is condensing, and this condense logarithms calculator does it with the law named at each step. Most a condensing logarithms calculator will hand you is the answer; this one shows which rule moved what, and stops when the bases do not match.
What Does Condensing a Logarithm Mean?
Condensing means writing a sum and difference of logarithms as one logarithm. The instruction on a worksheet is usually “write as a single logarithm” or “combine logarithms”, and both ask for the same thing.
It matters because a logarithmic equation cannot be solved while the logs are scattered. Once 2 log x + 3 log y − log z is a single log(x²y³/z), you can undo it in one move — raise the base to both sides and the log disappears. Scattered logs give you nothing to undo.
It appears in Algebra 2 and Pre-Calculus alongside expanding, and the two are always taught together because they are the same rules pointing in opposite directions.
The Three Laws This Condense Logarithms Calculator Uses
Three properties of logarithms do all the work, and they run in this order.
Power first. n log a = log(aⁿ). Every coefficient moves inside as an exponent. A fraction becomes a root: ½ log x is log√x, and ⅓ log x is the cube root.
Then product. log a + log b = log(ab). Adding logs multiplies what is inside them. It does not add them, which is worth saying out loud because it is the mistake almost everyone makes at least once.
Then quotient. log a − log b = log(a/b). Subtracting divides, so every term carrying a minus sign lands underneath the line.
Do them in that order and the answer falls out. Do the product before the power and the coefficients end up in the wrong place — which is the one thing that makes people think they cannot condense a logarithm at all.
Why Different Bases Cannot Be Joined
log x + ln y looks like it should condense. It does not, and no rearrangement helps.
All three laws assume one base. log means base 10 and ln means base e, so log x + ln y is a sum of two things measured on different scales — rather like adding a length in metres to one in feet without converting first. There is no single logarithm that equals it.
The calculator refuses this and tells you which two bases it found, instead of quietly producing something wrong. If you need them together, convert one with the change of base formula first, then condense.
How to Use the Condense Logarithms Calculator
Type the sum of logs and press =. The keypad has log, ln, the letters and a space, so 2 log x + 3 log y − log z is a handful of taps. Coefficients can be whole numbers, fractions like 1/2, or decimals like 0.5 — all three are read the same way.
Like terms are gathered before anything else, so 2 log x + 3 log x becomes log x⁵ rather than two separate pieces. If everything cancels the calculator says so plainly: log 1, which is 0 in every base.
One tap opens the laws that were used, and under that the working step by step. To go the other way, the expand logarithms calculator breaks a single log back apart — and running one after the other should land you exactly where you started. It is free to use online, with no account.
Condense Logarithms FAQ
How do you condense logarithms?
Does log a + log b equal log(a + b)?
Can you condense logs with different bases?
What happens to a fraction in front of a log?
What is the difference between condensing and expanding logarithms?
Coefficients up as powers, additions on top, subtractions underneath — and one base throughout. That is condensing, start to finish.