Expanding Logarithms Calculator
log(x³y/z) is one logarithm holding three things at once. Pulled apart it becomes 3 log x + log y − log z — three separate logs, each with a single letter in it.
That is all expanding is, and this expand logarithms calculator does it with the law named at every step, so the working reads the way your teacher wants it written. Most an expanding logarithms calculator will give you is the answer; this one shows which rule did what.
What Does Expanding a Logarithm Mean?
A logarithm can hold a whole expression inside it. Expanding means rewriting it as several logarithms added and subtracted, until nothing inside any of them can be broken down further.
The reason it is worth doing is not obvious from an algebra worksheet, so here it is: separate logs are far easier to work with. Differentiating log(x³y/z) directly is unpleasant; differentiating 3 log x + log y − log z is three easy pieces — which is the whole trick behind logarithmic differentiation in calculus. The same goes for solving equations: once the logs are separate you can gather them, cancel them, or take exponentials of both sides.
It turns up in Algebra 2, again in Pre-Calculus, and then properly in Calculus. The rules never change between those courses; only what you do with the result does.
The Three Laws This Expand Logarithms Calculator Uses
There are exactly three properties of logarithms that break an expression apart. These are the logarithm rules the whole topic runs on, and every question you will meet is some arrangement of them.
Product. log(ab) = log a + log b. Multiplication inside becomes addition outside.
Quotient. log(a/b) = log a − log b. Division becomes subtraction — and everything under the line takes a minus, which is where log(x/(yz)) trips people up. Both y and z come out negative.
Power. log(aⁿ) = n log a. The exponent moves out in front. A root counts as a fractional power, so log√x is ½ log x.
The calculator tags each step with the law it used, so a log expansion arrives with its reasoning attached rather than as a bare answer.
Why log(a + b) Does Not Expand
This is the one worth reading twice. There is no fourth law for a sum, and log(a + b) is not log a + log b.
Check it in one line. Put a = 1 and b = 1. The left side is log 2, which is about 0.301. The right side is log 1 + log 1, which is 0. They are not the same, and no amount of rearranging makes them the same.
So when a sum appears inside, it stays there. log((x+y)/z) becomes log(x+y) − log z — the quotient rule still applies, but the (x+y) is left exactly as it is. The calculator says so on screen rather than quietly inventing an expansion, because a wrong answer given confidently is worse than no answer.
How to Use the Expand Logarithms Calculator
Expanding logarithms by hand is quick once the rules are second nature, and this is the place to check the ones that are not yet. Type what sits inside the log and press =. The keypad has x, y, z, a and b, a root key, brackets and powers, which covers what these questions are made of. If you type the log as well — log(x³y/z) rather than x³y/z — it is understood and taken off first.
The answer appears as a sum of logs with exact coefficients: a half stays a half, never 0.5. One tap opens the laws that were used and why, and under that the full working step by step, ending with a check you can do yourself: put numbers in for the letters and both sides must agree.
If you want the value of a logarithm rather than its expansion, the log calculator handles any base. It is free to use online, with no account.
Expand Logarithms FAQ
How do you expand a logarithm?
Can you expand log(a + b)?
What is the difference between expanding and condensing logarithms?
Does the base matter when expanding logarithms?
What does it mean to expand a logarithmic expression?
Products to sums, quotients to differences, powers to the front — and a sum inside stays put. Those four sentences are the entire topic, whatever expand logarithms calculator you use to check your work.