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ACT Math: What’s a Logarithm?


If you find logarithms super-strange looking, you’re definitely not alone! They’re rarely tested on the ACT Math Test, but just in case you see 1 or 2 of these unusual questions, you’ll want to know the basic rules for solving them. We’re used to seeing exponents in a format like y = xa. In “logs” that equation is equal to logx(y) = a. Let’s look at an example with actual numbers:  32 = 9   is the equivalent of    log3(9) = 2

We would read the logarithm out loud as “log-base 3 of 9 equals 2.” A helpful way to remember this is to notice that whatever is on the other side of the equals sign is the exponent, and that the tiny number is the exponent base. Here’s a table with all the logarithm rules you might need to know on Test Day:


Translate the exponents to logs and the logs to exponents:


Now solve a few logarithms for the missing info:


As long as you know your exponent rules, you should have no problem with logarithms! Here’s a complete list of Logarithm rules (most of these will already be familiar to you from your study of exponents). The answers to these questions are:


One way to study these is to make flashcards with half of the equation of the front, and the other half on the back. Knowing these by sight will save you time on test day, since you won’t have to convert the logarithms to exponents first to solve.

Try another logarithm question on your own:


You could solve this by following rules for manipulating logorithms: log(a){(x/y)}^3 = 3 log(a) {(x/y)} =3 log(a) x - 3 log(a) y = 3m - 3n = 3(m - n)



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