The Natural Log

You are about to learn the single most important concept in solving exponential and logarithmic equations. The natural log is a very handy tool to keep in your mathematical tool belt in this chapter. The natural log is not only the inverse of the e^x function, but it is used directly in later sections to solve both exponential and logarithmic equations.

You will look at the graphs of the natural log function, practice using the properties, and also evaluate natural log functions on your calculator. Don't think too hard about making jokes about the natural log, we already know about them but we'll save you from the pain.

The natural log, ln, is a special case where you have f(x) = log_e x. The e is the base! If we wrote it out the long way, it would look like this: f(x) = log_{2.718281828...} x. In 1668, some bright guy named Nicholas Mercator decided to call it the natural log function. In the 1600's, we didn't have bloggers, we had loggers, also known as old fashioned nerds.

These are are equivalent statements:

f(x) = log_e x
f(x) = ln(x)

Notice how the base of the logarithm is e. We call this the natural log so we don't have to write log_e. We can simply write y = ln(x) and everybody understands what that stands for. Our little friend, e, wanted to have his own logarithmic function. What a guy!

Sample Problem

Graph the natural log function and it's inverse.

The blue curve is the y = e^x function, and the natural log function y = ln(x) is the red curve. Since they are inverses, they are basically a reflection over that imaginary diagonal line y = x. 

Sample Problem

Using the properties of logarithms (from our earlier section), rewrite this natural log function as a sum or a difference of logarithms:

Rewrite  in sums and differences of natural logarithms.

Okay let's do this in steps, first since it's a natural log, it follows the same properties of logarithms (which we learned in an earlier section).

Speaking of inverses, did you know that we could apply the inverse of the natural log to the exponential function? Watch this magic:

e^lnx = x
ln(e^x) = x

Sample Problem

Evaluate e^ln9.

Since we now know that the inverse of an e function is ln, the solution is:

e^ln9 = 9.

Sample Problem

Evaluate ln(e^1.9).

Easy Shmeazy. The answer is 1.9 thanks to our trusty friend, the inverse.

In the last section, Graphs of Logarithmic Functions, we mentioned that you could convert a logarithm of a certain base to a logarithm of base 10 with a simple trick.

Remember this...

y = log₂ x

...is the same as this:

Similarly, we can do this with natural logs which will help us in the next section solve equations. If we have y = log₃ x, we can convert it to a natural log function, which some people seem to like better than working with logs.

Natural logs are less cumbersome and there are nifty little buttons on your calculator that are for ln and you don't have to worry about changing bases. Break out the granola, we're going natural!

Sample Problem

Convert log₆ 100 to natural logs and then evaluate (round to nearest hundredth).