The natural logarithm

There are a few numbers in mathematics which are more beloved than others. One is the number , also called Euler's number.

An exponential function looks like this:

The most useful one is , and we refer to it as the exponential function. It appears everywhere when talking about natural growth. Sometimes, you will see it denoted as .

How can we define this function? The number , but the decimals actually continue forever so it doesn't feel like should be very pretty. It turns out to be quite neat, though.

Let's look at an example. Say that your bank offers an increase by times the money if you keep them in the bank for a year.

Assume further that keeping the money for half a year in the bank gives you an interest of , and so on for shorter periods.

Then, keeping the money in for a year would give you:

times what you put in. Taking the money out half way through the year, and putting the new amount straight back in would give you:

times the original amount. That's a lot more!

If you choose to take the money out and put it back every day of the year, you'd have this many times the money by the end of the year:

We call this a binomial sum. Writing all the terms out, one has to multiply each and every term from each of the parentheses together. This way, you end up with a ton of extra stuff, compared to if you only took the money out two or three times.

Making the time periods shorter still, you'd end up with:

times the original amount. As you may have guessed, this adds still more terms to the sum, which thus keeps growing. Finally, taking the limit of the above expression, results in our definition of :

A side note: the bank system has a built-in protection against the type of trading described above. So, for getting rich, this method is not recommended. However, exponential growth (with different constants in front) realistically describes phenomena spanning bacterial growth, processing power of computers, nuclear reactions and much more.

The exponential function is particularly interesting because of some properties that will be clarified later in this course. But sometimes we also need exponential functions with another basis :

We will dedicate this section to stating some rules for this type of function. They may seem numerous, but they are very useful, so give them a moment of consideration before moving on.

The exponential functions can be separated into two categories:

The first type with and can be seen below:

Functions with a base go to infinity as goes to , and to as goes to infinity.

If we instead let and , we get this shape:

If , the function goes to infinity as goes to infinity, and to as goes to negative infinity.

No matter the base, all exponential functions follow the same pattern with regards to some rules.

The following properties relate to different values of the exponent:

1)

2)

3)

4)

The second rule implies that all exponential graphs will pass through the point .

Let and , and let and be real numbers. Then, the following rules apply:

With these in your pocket, you are ready to handle exponential calculations.

As you're about to eat, you discover that your pasta carbonara has been ruined. It's all covered with mold. Yet, you only kept it in the fridge for a couple of days!

Mold grows fast. If you start out with one lump of mold, you might have something like two lumps of mold next day. On the following day, both lumps of mold have grown, and that starts a vicious cycle.

If the area of mold increases by a factor of every day, the total area can be computed as

Here, is the number of days.

is your lucky number, so you try computing when the mold will cover cm. Thus, you run into the equation:

You sigh in despair. How should you solve this kind of equation? When you plug in the equation into Wolfram Alpha, you get:

What does the mean here? The logarithm of , denoted , answers the question:

- what power of equals ? It usually can't be computed by hand.

The logarithm can also be thought of as the inverse function of the exponential function . If:

then:

In general, we require that and .

Here are the graphs of and :

There are a handful of logarithm rules, all of which are worth memorizing. Here they are in all their glory:

As you're reading this, you suddenly feel like having some Earl Grey tea. After boiling the water and pouring it into the cup, you go off to continue reading.

You'll have to wait a while for the water to cool off, or else you'll burn your tongue. Remember, once you tried drinking the tea shortly after you'd boiled the water and your tongue became all covered by white blisters...

This time, you want to be sure the tea isn't too warm. A quick Google search reveals that the ideal tea drinking temperature is 57°C. But for how long should you wait?

The temperature decreases quicker when the tea is hot. You get something like this:

Here, is the time that has passed since you poured in the water. is the temperature of the room, around °C.

As it turns out, this relation is satisfied if:

where and are constants. Since , . For simplicity, we'll assume the other constant is just . This is all we need to compute the time.

Ok, here we're stuck. We need to use something else to solve this kind of equation: namely, the natural logarithm.

The natural logarithm of a number , written , answers the question to the what, equals ? Hence, . It's also the inverse function of .

Back to our example then:

We end up with hours, or about minutes. Our choice of the constant, , probably meant that our cup had some isolating coating.

Here are the graphs of and :

The natural logarithm allows us to solve an array of new problems. With the natural logarithm, we can also rewrite any exponential function, like this:

One of the many practical uses of the exponential function is to define a special group of functions, known as the hyperbolic functions:

The names of the functions, as well as the relationships between them, hints at the similarities to the trigonometric functions.

The truth is that despite looking rather different at first, these functions have a lot in common with their respective trigonometric counterpart.

From our definitions, proving the following hyperbolic identity is trivial:

Proof:

The equation looks similar to the trigonometric identity:

However, the negative sign between the two expression makes the unit hyperbola the object connecting and , rather than the unit circle as for and

The graphs of the three hyperbolic functions we have defined look as follows:

These functions occur naturally in various situations around us. For instance, a uniform freely hanging wire attached in two ends, will follow the curve of a hyperbolic cosine function.

Imagine a clothesline. Right after washing, the wet heavy clothes will weigh down the line and disturb its shape. As the clothes dry and get lighter though, the line will resemble a hyperbolic cosine function better and better.