In earlier courses, you may have seen logarithms defined in terms of raising bases to powers. For example,

In those terms, the * natural logarithm* should be the power to which you raise e to
get x. (Remember that is just shorthand for .)

Now

e is represented by an infinite non-repeating decimal (like ). So, for instance, is the power to which you raise to get 4. How would you figure that out?

You might also wonder where e comes from: How do you compute it? And why choose such an ugly number for a logarithm base?

In this section, I'll take a different approach to the natural log. I'll define it using calculus as the area under a curve. For starters, this allows us to compute its derivative easily. But what does this have to do with logarithms defined in terms of raising bases to powers?

When you studied logarithms, you learned that they satisfy certain properties:

(a) .

(b) .

(c) .

(d) .

I'll show that the log I define as an area satisfies those properties. That gives some justification for considering it to be a "logarithm".

To begin with, the Power Rule says

The formula does not apply to

An antiderivative of would have to satisfy

But the Fundamental Theorem implies that if , then

Thus, plays the role of .

Define the * natural log* function by

By construction, if ,

represents the area under from 1 to x:

has many properties you'd expect a logarithm to have. For example,

You'd expect the log of a product to equal the sum of the logs. If a and b are positive numbers, then

In the second integral, let , so , and . When , ; when , . So

In other words,

In similar fashion, you can verify that

Thus, there is some justification in calling a logarithm, because it has the same properties you'd expect logs to have.

Here are some additional properties of .

First,

Therefore, the graph of is increasing for .

Moreover,

Therefore, the graph of is concave down for .

Next, consider the following picture:

The area under the curve from 1 to 4 is . It is greater than the sum of the areas of the three rectangles, so

If n is a positive integer, then

So if , then

Since n is an arbitrary positive integer, I can make arbitrarily large by making x sufficiently large. This proves that

Here's the graph of :

The differentiation formula for works together with the other differentiation rules in the usual ways.

* Example.* Compute:

(a) .

(b) .

(c) .

(d) .

(e) .

(a)

(b)

(c)

(d)

(e)

If I say that " " --- the
derivative of is --- then should be defined
wherever is defined. Therefore, it is not really
correct to say *without the qualification* that " ". For is defined
for , whereas is only defined for .

It turns out that the correct statement is:

For , this is the same as the old formula. For , , so

So our new antiderivative formula is

You can omit the absolute value signs if the quantity inside is never negative. For example,

Since is always positive, I can write " " instead of " ".

* Example.* Compute .

* Example.* Compute .

You can use * logarithmic differentiation* to
compute derivatives which are difficult to compute in other ways.

* Example.* Compute .

Let . Taking logs and bringing the power down, I get

Differentiate both sides, using the Chain Rule on the left and the Product Rule (and Chain Rule) on the right:

Multiply both sides by y to clear the fraction on the left, then substitute :

I can use calculus to construct the * exponential
function* .

For , I have

Since , I have . This means that increases for . Next, I use following facts:

(a) An increasing function f is * injective*: If
, then .

(b) An injective function has an inverse function .

Applying these facts to , I find that it has an inverse function, which I will denote (or ). Thus:

1. is a function from to the positive real numbers (because is a function from the positive real numbers to ).

2. for and for .

It's easy to use the properties of the natural log to derive corresponding properties of :

(a) .

(b) .

(c) .

(d) .

To complete the discussion, I can use and to define logs and exponentials to other bases.

(a) If and , define .

(b) If , define .

With these definitions,

In words, this says that is the power to which you raise a to get x --- the definition of a log in terms of a base raised to a power that you knew before.

Using the definitions for and , I also get the differentiation formulas

The point of doing things using calculus is not that we have any new
properties. Rather, it allows us to derive properties *which were
just stated and taken for granted* in a rigorous way.

Copyright 2018 by Bruce Ikenaga