Logarithms and Exponential Functions
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 = 10x
log10y = 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 log10 y?The exponent. Specifically, it is the exponent you need to raise the base 10 to so you get y. For example, what is log10100? 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 = 10x and log10 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.
What is log4 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. 43 = 64, so log464 = 3.
You may have noticed before that we used a logarithm to simplify the following equation:
log10y = log1010x
log10y = x
log1010x = x because the exponent you need to raise 10 to so you get 10xis 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.
What is the inverse of y = 42x?
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 = 42x
log4y = log442x
log4 y = 2x