# 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* = 10^{x}

log_{10 }*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 log_{10} *y* is the exponent you need to raise the base 10 to so you get *y*. For example, what is log_{10} 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 10^{2} = 100.

log_{10} 100 = 2

Logarithms are just another way of writing down exponents. The equations *y* = 10^{x} and log_{10} *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 log_{4} 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.

4^{1} = 4

4^{2} = 16

4^{3} = 64

Ah, there we go: 4^{3} = 64, so log_{4 }64 = 3.

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

log_{10 }*y = *log_{10 }10^{x}

log_{10 }*y* = *x*

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

log* _{b} b^{x}* =

*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:

*b*^{logb 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* = log_{10 }*x*

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

### Sample Problem

Solve *y* = 4^{2x} 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* = 4^{2x}

log_{4 }*y* = log_{4 }4^{2x}

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 4^{2x} ?" The answer is just 2*x*. In other words, the log_{4} cancels out the 4 and leaves us with the exponent.

log_{4} *y* = 2*x*