- Topics At a Glance
**Linking Exponents and Logarithms****Inverse Functions**- Rules for Inverse Functions
- The Base
- The Natural Log
- Exponential Functions
- Linear and Exponential Growth
- Exponential Growth and Decay
- Solving Exponential Equations
- Limits of Exponential Functions
- Logarithmic Functions
- Revisiting Inverse Operations
- Change of Base
- Limits of Logarithmic Functions
- Properties of Exponents and Logarithms
- In the Real World

First and foremost, you've got to know that logarithms are *inverse operations* of exponentials. Yup, we're turning your world upside-down. Please rotate your monitor around so that you can read the rest of this section. If you can't, you'll just have to get used to reading everything that way. Unfortunately we've run out of hot towels for all the sore necks caused by this new policy, you're going to have to provide those yourself.

Mathematics gets stuff done through operations. Otherwise we'd have a bunch of values and nothing to do with them; we'd have no way to connect them. Enter operations. Operations string values together to produce other values. However, sometimes we need stuff undone, so we created inverse operations. Inverse operations are mathematical takesy backsies, they are operations that undo one another. Addition and subtraction are inverse operations of each other. Multiplication and division are also inverse operations of each other. If you started with 3, multiplied it times 2, but then divided the answer by 2, you'd be right back to 3.

Guess where we're going with this? Exponents and logarithms are inverse operations! So if we have this equation:

*y* = *x*

We can add to both sides… *y* + 2 = *x* + 2

We can subtract to both sides… *y* – 2 = *x* – 2

Multiply… 2*y* = 2*x*

Divide…

You can *also* "**exponentiate**" both sides: 10^{y} = 10^{x}

And finally "**take the log**" of both sides: log *y* = log *x*.

Every single one of the expressions here are equivalent to *y* = *x* and to each other. Because you know that logs and exponents are inverse operations, just like addition and subtraction are, manipulating equations with exponents will be a snap.

Now that you know exponentiating and taking the log are inverse operations, let's talk inverse functions.

Imagine a function that has this set of inputs and outputs. No really, close your eyes and focus on your inner mathlete:

{(1, 5), (2, 9), (3, 13), (4, 17)}

If you're on the up and up, you might recognize these numbers come from the linear function *y* = 4*x* + 1. Awesome. The inverse of this function simply switches all of the input and output values:

{(5, 1), (9, 2), (13, 3), (17, 4)}

What's that function? No cheating, we've got robots working for us.

A very simple way of figuring out the answer to this question is to solve for *x* and switch the *x* and *y*.

*y* = 4*x* + 1*y* *–* 1 = 4*x*

Then, we switch the *x* and the *y*

Bam! There's your answer. is the inverse of* y = 4x + 1. *A clearer way to represent this would be to say .

Special note: *f*^{ -1}(*x*), the inverse function, is **not** the same as , the reciprocal function. Be careful!

If you applied a function and its inverse function to an input value, the input would come out of the works unchanged. Prove this to yourself: try throwing a 4 into either function and then put the answer in the other.

See? It's still four. (If not, something has gone terribly wrong and this message will self-destruct in 3 seconds.) In other, more math-y, words: *f*^{ -1}[*f*(*x*)] = *x*.

Here's some proof:

*Step 1, The Awesome Step: y = 4( 4) +1 = 17Step 2, The Awesomer Step: *

Impressed, yet? It's like a magician who walks on the stage, pulls a rabbit out of the hat, and…*WHOA THAT'S NOT A RABBIT! *He quickly puts the non-rabbit back into the hat and walks backwards off the stage as if nothing happened. No refunds.

In the next section we'll try to find the inverse of an exponential function. It's not going to be quite as easy as this section, but you'll feel even awesomer-er when you figure things out.

Apply the magical lesson of the inverse to an exponential function. First, take a look at an exponential function. Like we mentioned before, exponential functions are simpler than they look: just stick the *x* in the exponent.

Here's one:

*y* = 10^{x}

Right. So what about its inverse? Just solve for *x*. Wait—how is this supposed to work?

Without logarithms, you might be stuck in the mud. Dirty, goopy, nasty mud that won't wash out no matter how many times you put it in the washer. Math can do anything; it can even keep your clothes clean and fresh. (Disclaimer: Do not attempt to put math in the washer, this will void all warranties.)

Go along with us for a second here, we're going to use a *logarithm* to solve for *x* and then switch things around. Like we said in the last section, taking the log is the inverse operation of exponentiation, so we can use it to cancel the 10 on the right side of the equation:

*y *= 10^{x} × log_{10}

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

log_{10}* y* = *x*

log_{10 }*x* = *y*

*y* = log_{10 }*x*

Ta-daaa! *y* = log_{10}* x*, a logarithmic function, is the inverse of *y* = 10^{x}, an exponential function*. *