ACT Math: A Lesson on Logarithms

What are logs in math? Do you use them to develop a foundation and build like beavers do? No, you don’t. Instead, in math, logs are the “opposite” of exponentials, just as subtraction is the opposite of addition. If I asked you what number (x) to the third power equals 8 (x³ = 8), then you would take the cube root of both sides and tell me the cube root of 8 equals 2.

Now consider this: if I asked you 2 raised to what power (x) gets you 8, how would you solve it? Well, we know that x=3 because 2³ = 8, as we saw from the previous problem. But what steps would you take to solve this problem, or any others like it? As I mentioned before, logs are the “opposite”, or the inverse, of exponentials. Thus, one operation can undo the other. Let’s take a look at the relationship between them.

Relationship between Exponentials and Logs

y=b^x           log_b(y)=x

Example

8=2^x           log₂(8)=x ; x=3

Just as in the exponential, the base (b) is always positive and never equal to one. In both cases, a helpful way to remember the relationship is (b) is the base in both scenarios. The x and the y switch, however, as can be seen first by the (x) on the same side as the (b), and then on the same side as the (y). Another easy way to remember this is that whatever the argument of the logarithmic expression is, in this case (y), becomes the “equals to” in the exponential expression.

Unless the base (b) is equal to 10, you will not be able to just plug the expression into your calculator to evaluate that logarithmic expression. Your calculator will only evaluate logs in base 10. However, do not fear! You can easily use change of base to change the logarithmic expressions to base 10. Here is how you can do that!

Change of Base Property

log_b(y)= log_a(y)/ log_a(b)

Example

log₂(8) = log₁₀ (8)/log₁₀(2)

you can enter this in your calculator as log(8)/log(2)

There are also two key properties to know regarding adding and subtracting the arguments of logs!

Addition/Subtraction Properties of Logs

log(xy)= log(x)+log(y)
log(x/y)= log(x)-log(y)

With this new understanding of logs, you have the basic tools to face the logarithm questions on the ACT! Be sure to always pace yourself, be mindful and breathe.

You’ll do great.