exponential and logarithmic functions

I am working on converting logarithmic equations in exponential form but I do not understand how my book explains it.

The two problems I am having trouble with are:

ln e^3 = 3

ln 1/e^2 = -2

Could someone please explain how to do this so I understand the concept?

Suppose you have an exponential equation like:
$a^b = c$

The "logarithm" function says: "What number do I have to raise $a$ to so as to get $c$ ?"

So we have $\log_a c = b$ .

Now when the number being raised to the power is $e$ (which is of course 2.71828 ...), the same thing applies.

So $e^x = y$ is another way of saying $\log_e y = x$ . And a special symbol for $\log_e$ is $\ln$ .

Let's look at $\ln e^3 = x$ , where we're trying to find $x$ .

This is saying: "What number do I raise $e$ to so as to get $e^3$ ?"

What number do I raise $n$ to (where $n$ is any number) to get $n^3$ ?

As you can see, $\ln{}$ "undoes" the work that " $e$ to the power of" does.

That is, logarithm and exponential (if it's the same base) are inverse functions.

Now the second one is trickier.

$\ln \left({\frac 1 {e^2}}\right) = x$

The thing here is to remember that $\ln \left({\frac 1 {x}}\right) = - \ln x$ (it just is, the book probably shows why).

So $\ln \left({\frac 1 {e^2}}\right) = -\ln e^2 = -2$ (from what we did before).

Thanks

Wow!

Thank you very much, that explanation was so much better than my book.

I appreciate it.