The Expanding Logarithms Calculator
Welcome to the expanding logarithms calculator, where we'll learn to expand logarithmic expressions according to three simple formulas. The first one, the product property of logarithms, basically turns multiplication inside a log into adding logs. The formula for division works the same, but the sum changes into a difference. Lastly, the power property of logarithms allows us to take the exponent outside.
Arguably, the above description was vague. Hopefully, vague enough to inspire curiosity and willingness to see the log addition, difference, and exponent rules in detail and study how to expand the logarithmic expression together with our expand log calculator.
What is a logarithm?
Some of you may disagree, but deep down, we all know that mathematics lies at the bottom of it all. From the very moment two rabbits met each other and decided to prolong the existence of their species, all they had in mind was mathematics, even if they didn't quite know it.
Apart from raising the probability of their survival (which, as we know, was very successful), the rabbits managed to become a symbol of multiplication. The meeting of two such animals leads to more rabbits. Therefore, if, for instance, we had 4 couples and each got 6 younglings, we describe the overall number of little ones using multiplication:
Obviously, the next generation has similar interests to their parents. Therefore, if each of a couple's kids got another 6 small ones later in life, and the same for their kids, and their kids, then we can calculate how big the fourth generation is with a new mathematical tool — exponents:
The logarithm is the inverse operation to an exponent. Our animal example corresponds to wondering how many generations have to pass before we get a fixed number of little ones. Therefore, for instance:
Observe that the little 6 corresponds to the size of a single litter. Note that taking a root is also considered an inverse operation to taking a power. However, as opposed to logarithms, roots return the exponent base, not the exponent itself — in the above language, they return the size of a single litter, not the number of generations that have to pass. For example:
And for those who prefer symbols instead of bunnies, here's the formal definition of a logarithm:
loga(b) gives you the power to which you'd need to raise a in order to obtain b.
Before we expand the logarithmic expressions and see the formulas, let's mention a few essential facts concerning logs:
- There are two very special cases of the logarithm which have unique notation: the natural logarithm and the logarithm with base 10. We denote them
ln(x)andlog(x)(the second one simply without the small 10), and their bases are, respectively, the Euler number e and (surprise, surprise!) the number 10. There is also the binary logarithm, i.e., log with base 2, but it's not as common as the first two. - The logarithm function is defined only for positive numbers. In other words, whenever we write
loga(b), we require b to be positive. - Whatever the base, the logarithm of 1 is equal to 0. After all, whatever we raise to power 0, we get 1.
- Logarithms are extremely important. And we mean EXTREMELY important. Outside of mathematics, they're used in statistics (e.g., the lognormal distribution), economics (e.g., the GDP index), medicine (e.g., the QUICKI index), and chemistry (e.g., the half-life decay). Also, quite a few physical units are based on logarithms — for instance, the Richter scale, the pH scale, and the dB scale.
Alright, we've come to know our new friend reasonably well. It's finally time to see how to expand logarithms, and we begin with the product property of logarithms, also known as the log addition rules.
Product property of logarithms: log addition rules
The logarithm of a product is a sum of logarithms.
Why should that be true? Because a logarithm is an exponent, and exponents already add when you multiply powers: bm · bn = bm+n. Write x and y as powers of the base, multiply them, and the exponents add — the product rule is that fact read backwards.
For example, in base 10: log(4 · 25) = log(4) + log(25) = 0.60206 + 1.39794 = 2, which checks out, since 4 · 25 = 100 and log(100) = 2.
Quotient property of logarithms: log difference rules
The logarithm of a quotient is a difference of logarithms.
For example, in base 10: log(200 / 2) = log(200) − log(2) = 2.30103 − 0.30103 = 2, and indeed 200 / 2 = 100.
Here's something worth noticing, and it's the idea this calculator is built around: the quotient rule is not really a separate law at all. Since x / y = x · y−1, dividing is just multiplying by a negative power. Apply the product rule and then the power rule, and the difference appears on its own. That is why our calculator asks you to place each factor in the numerator or the denominator: a denominator factor is simply a factor whose exponent carries a minus sign.
Power property of logarithms: log exponent rules
The logarithm of a power is the exponent times the logarithm.
For a whole number n, this is just the product rule applied over and over: x3 = x · x · x, so log(x³) = log(x) + log(x) + log(x) = 3·log(x). It then extends to fractional and negative exponents by continuity, which is what lets you pull a square root out of a log as a factor of ½.
The three rules are really one
Once you allow negative exponents, all three properties collapse into a single formula:
Set every exponent to 1 and you get the product rule. Use a single factor and you get the power rule. Make one exponent negative and you get the quotient rule. Everything this calculator prints is that one identity in action.
Example: using the expanding logarithms calculator
Suppose we want to expand log₁₀( 100² · 1000 / 10 ) — the calculator's default. Enter base 10, then the factors: 100 with exponent 2 in the numerator, 1000 with exponent 1 in the numerator, and 10 with exponent 1 in the denominator. The calculator works through it like this:
- Quotient rule: log₁₀(100² · 1000) − log₁₀(10)
- Product rule: log₁₀(100²) + log₁₀(1000) − log₁₀(10)
- Power rule: 2·log₁₀(100) + log₁₀(1000) − log₁₀(10)
- Evaluate: 2·2 + 3 − 1 = 6
And the check works: the argument is 100² · 1000 / 10 = 1,000,000 = 10⁶, so its log base 10 really is 6. The result panel verifies this for you both ways — via the expanded sum and via the product — and reports the difference between them.
How far can you expand? All the way to the primes
Here's a question most textbooks skip: when is an expansion actually finished? The answer is that expansion has a floor, and the floor is prime. As long as an argument is composite, the product rule can still split it further. In our example, 100 = 2²·5² and 1000 = 2³·5³ and 10 = 2·5, so we can keep going:
And it still comes to 6, because log₁₀(2) + log₁₀(5) = log₁₀(10) = 1. The calculator computes this fully expanded form for you whenever your arguments are whole numbers — and it will tell you when primes cancel out of the expression entirely.
Why expanding is more than a homework exercise
Expanding a logarithm is not just tidying. It is often the only way to get an answer at all. A product of many large factors can easily overflow what a computer can represent, but the sum of their logarithms stays perfectly manageable. Enter a factor like 10 with exponent 400 and you can watch it happen: the product blows past double precision, while the expanded sum calmly reports 400. Turning products too big to hold into sums you can still add is precisely why logarithms were invented.
US and metric units, and money
The optional unit panels let you read the argument X as a physical quantity in either the metric (SI) system or the US system, and they demonstrate a small but important fact: changing units only adds a constant. Since log(k·X) = log(k) + log(X), rescaling shifts every logarithm by the same amount and leaves the shape of the expansion untouched — which is exactly why a log-log plot keeps the same slope no matter which units you measure in.
The money panel puts the quotient rule to work on a real question: how long does it take to grow a principal P into a target X at rate r? The answer, n = [log(X) − log(P)] / log(1+r), is the quotient rule turning the ratio X/P into a difference you can actually compute. You can pick your currency from twenty world currencies, with the US dollar and the Russian ruble at the top of the list (EUR, GBP, JPY, CNY, CAD, AUD, CHF, INR, BRL, MXN, KRW, SEK, NOK, DKK, PLN, CZK, HUF, TRY).
Frequently Asked Questions
What does it mean to expand a logarithm?
Expanding means rewriting a single logarithm whose argument is a product, quotient, or power as a sum and difference of simpler logarithms — using the product, quotient, and power properties of logarithms.
Is log(x + y) the same as log(x) + log(y)?
No — and this is the single most common mistake with logarithms. The rules apply to products, never to sums. There is no rule that simplifies log(x + y). What is true is the other way round: log(x) + log(y) = log(x · y). The calculator shows you both numbers side by side so you can see how far apart they really are.
Can I expand logs with different bases?
The rules apply within one base at a time. If you need to change base, use logb(x) = ln(x) / ln(b). Note that expanding gives the same shape in every base — only the numerical values scale.
How do I pull an exponent out of a logarithm?
Use the power rule: an exponent on the argument becomes a coefficient in front of the log, so log(x³) becomes 3·log(x). A square root is the exponent ½, so log(√x) = ½·log(x).
Why must the argument be positive?
Because a logarithm asks "to what power do we raise the base?" — and raising a positive base to any real power always gives a positive result. No exponent can ever produce zero or a negative number, so the log of such a value doesn't exist among the real numbers.
Is expanding the opposite of condensing logarithms?
Yes. Condensing collapses a sum and difference of logs into one logarithm; expanding breaks a single logarithm apart. Both use the same three rules, just in opposite directions.
References
- Weisstein, Eric W. "Logarithm." MathWorld — A Wolfram Web Resource.
- Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, Chapter 4: Elementary Transcendental Functions.