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# Logarithms and Exponential Functions

# The Base

Now that we know about how exponents and logarithms are inverse operations of each other, and logarithmic functions are the inverse of exponential functions, 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?

Well, you can still see the *y*; you can see the 10; the only thing that's left is the *x*. Bingo. It's one in the same. What is log_{10} *y*?The **exponent**. Specifically, it 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. Logarithms are another way of writing down exponents. *y* = 10^{x} and log_{10} *y* = *x *are completely equivalent to each other.

We mentioned earlier that exponential and logarithmic functions and operations are inverses of one another. That's still true, but only the functions and operations are inverses. See the illustration below, and this state makes a little more sense.

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 is 4. So, what is the exponent needed to give 32? Not 2, that yields 16. It's 3. 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

log_{10}10^{x }= x because the exponent you need to raise 10 to so you get 10^{x}is x. This can be used for any other base. A log can have a base of any positive number, and 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. You might see a logarithm 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.

### Sample Problem

What is the inverse of y = 4^{2x}?

Now that we know all logs have a base, and they can have a base of any positive number, let's use one to invert this exponential function.

y = 4^{2x}

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

log_{4} y = 2x