The Inverse Properties of Logarithms

Here are two properties of logarithms that are sometimes called the inverse properties.

log_aa^x=x
a^loga^x=x

They are called this, because they come from the fact that the log functions are inverses of the exponential functions. To get these properties from that consideration let f(x)=a^x. Then f^-1(x)=log_ax. The fundamental rule about inverse functions in general is that f(f^-1(x))=x and f(f^-1(x))=x. Applied to our f here this gives precisely these two properties above, since in our case f(f^-1(x)=f(log_ax)=a^loga^x and f^-1(f(x))=log_aa^x.

But whenever I see this justification it always sounds unnecessarily fancy, because you shouldn't really need to talk about functions in order to understand this. There is another way to understand this that I think is helpful for students in understanding the meaning of logarithms that doesn't need to have anything to do with functions and their inverses. To think about it this way instead of thinking of log as the inverse function of exponential you think about it like this.

log_ax means the power that you raise a to in order to get x

Do a few examples to get comfortable with the matter.

Example 1

log₂8

To find this we are looking for a power that we can raise 2 to in order to get 8. That would be 3, so 3 is the answer.

Example 2

log₁₀10,000

What power do we raise 10 to in order to get 10,000. Just count up the zeros to get the answer is 4.

Example 3

log₄8

This is asking what power can you raise 4 to in order to get 8. This is a little trickier, because there is no amount of times that you can multiply 4 times itself to get 8, but if we think about the meaning of fractional exponents then we can figure it out. Fractional exponents have to do with roots and the square root of 4 is 2. I can find a number I can raise 2 to in order to get 8. That's the 3 from Example 1. Putting these together we can see that if we raise raise 4 to the 3/2 power that will give us 8, because raising to the 3/2 power is taking the square root, and then raising to the 3rd power, so 3/2 is the answer to this one.

Now after getting comfortable with this definition, back to the inverse properties. Look at the first one again.

log_aa^x=x

The left side of it asks what power can you raise a to in order to get a^x. In other words a^?=a^x. Now that's pretty easy, almost easy enough to make it difficult. The answer must be x.

Now look at the second one.

a^loga^x=x

The exponent here log_ax represents the power that you raise a to in order to get x, so what happens when you raise a to that power (which we are doing here), well, I just told you, that log_ax was precisely chosen to be a power that you can raise a to in order to get x, so if you raise a to it you had better get x or it isn't the right answer to the log problem.

Admittedly it takes a bit of thinking these through to understand them this way, but I think when you do you will find it rewarding, because if you think this reasoning through properly these identities will become so obvious that you couldn't possibly forget them any more than you could forget that Grant was buried in Grant's tomb.