# The Base

## Translating Logs

Now that we know about how exponents and logarithms are inverse operations of each other, it's time to explore the deep, dark caverns of the base. Take a look at the function we inverted before:

y = 10x
log10 y = x

The first thing to pop out like a jack-in-a-box is that 10, the base of the exponent, is now attached below the log. Guess what? It's also called the base: the base of the logarithm.

So where's the exponent?

It's still the x. Bingo. It's one and the same.

That means the entire expression log10 y is the exponent you need to raise the base 10 to so you get y. For example, what is log10 100? To figure this out, ask yourself the following question: 10 raised to the power of what would give 100? The answer is 2, because 102 = 100.

log10 100 = 2

Logarithms are just another way of writing down exponents. The equations y = 10x and log10 y = x are completely equivalent to each other.

Here's an illustration showing how logs and exponents are related:

In fact, we can represent any exponential in log form and vice versa.

### Sample Problem

What is log4 64?

If you look at the base, it's 4. So, what is the exponent needed to give us an answer of 64? Let's try sticking a few exponents on a base of 4.

41 = 4
42 = 16
43 = 64

Ah, there we go: 43 = 64, so log64 = 3.

You may have noticed before that we used a logarithm to simplify the following equation:

log10 y = log10 10x
log10 y = x

The reason log1010x = x is because the exponent you need to raise 10 to so you get 10x is x. This can be used for any other base:

logb bx = x

And it works the other way, too. If we have a number raised to a log whose base is that same number, they cancel out:

blogb x = x

A log can have a base of any positive number, any one you like. It's also possible to have a log with a negative base, but they're mean, nasty, and don't even help clean up after eating dinner. You can safely avoid them.

Logarithms always have a base, just like those couples that just can't get away from each other. If you see a logarithm written without a base, don't panic. It's not lonely; the base just has an invisibility cloak. Which base does it have? Good ol' number 10.

log x = log10 x

A base-10 log even has a special name: the common logarithm.

### Sample Problem

Solve y = 42x for x.

Now that we know all logs have a base, and they can have a base of any positive number, let's throw a log with a base of 4 on both sides of our equation.

y = 42x

logy = log42x

The stuff on the right side is asking, "What exponent do we need to stick on a base of 4 to get an answer of 42x ?" The answer is just 2x. In other words, the log4 cancels out the 4 and leaves us with the exponent.

log4 y = 2x