All AboUt lEsSon: Summary of "chapter 4" (1)

Exponential Functions

As we have seen, exponential functions describe events that grow (or decline) at a constant percent rate, such as placing capital in a savings account. Exponential functions will have a role to play in borrowing and lending; investing; discounting; and growth of companies. Many of these equations need additional mathematics topics such as series and sequences learned later in mathematics.

Rules of Exponents:

If b is any number such that

and

then an exponential function is a function in the form,

where b is called the base and x can be any real number.

Before we get too far into this section we should address the restrictions on b. We avoid one and zero because in this case the function would be,

and these are constant functions and won’t have many of the same properties that general exponential functions have.

Next, we avoid negative numbers so that we don’t get any complex values out of the function evaluation. For instance if we allowed

the function would be,

and as you can see there are some function evaluations that will give complex numbers. We only want real numbers to arise from function evaluation and so to make sure of this we require that b not be a negative number.

Example 1 Sketch the graph of

and

on the same axis system.

Solution

Okay, since we don’t have any knowledge on what these graphs look like we’re going to have to pick some values of xand do some function evaluations. Function evaluation with exponential functions works in exactly the same manner that all function evaluation has worked to this point. Whatever is in the parenthesis on the left we substitute into all the x’s on the right side.

Here are some evaluations for these two functions,

x
-2
-1
0
1
2

Here is the sketch of the two graphs.

Properties of

The graph of

will always contain the point

. Or put another way,

regardless of the value of b.

For every possible b

. Note that this implies that

then the graph of

will decrease as we move from left to right. Check out the graph of

above for verification of this property.

then the graph of

will increase as we move from left to right. Check out the graph of

above for verification of this property.

then

As a final topic in this section we need to discuss a special exponential function. In fact this is so special that for many people this is the exponential function. Here it is,natural exponential function.

where

. Note the difference between

and

. In the first case b is any number that is meets the restrictions given above while e is a very specific number. Also note that e is not a terminating decimal.

This function is simply a "version" of

where b >1.

Logarithm Functions

Here is the definition of the logarithm function.

If b is any number such that

and

then,

We usually read this as “log base b of x”.

Note that the requirement that

is really a result of the fact that we are also requiring

. If you think about it, it will make sense. We are raising a positive number to an exponent and so there is no way that the result can possible be anything other than another positive number. It is very important to remember that we can’t take the logarithm of zero or a negative number.

Let's examine the function

Check out the table and the graph. Remember that x must be positive.

x	f (x) = y
1/4
1/2
1
2
3
4

Natural Logarithmic Function:

The function defined by

is called the natural logarithmic function.

(e is an irrational number, approximately 2.71828183, named after the 18th century Swiss mathematician, Leonhard Euler .)

Notice how the characteristics of this graph are similar to those seen above.

This function is simply a "version" of

where b >1.

Inverse of

: Since

is a one-to-one function, we know that its inverse will also be a function.
When we graph the inverse of the natural logarithmic function, we notice that we obtain the natural exponential function, f (x) = e^x.
Notice how (1,0) from y = ln x becomes (0,1) for f (x) = e^x. The coordinates switch places between a graph and its inverse.

Example 1 Evaluate each of the following logarithms.

(a)

(b)

Solution :

(a)

Okay what we are really asking here is the following.

As suggested above, let’s convert this to exponential form.

Most people cannot evaluate the logarithm

right off the top of their head. However, most people can determine the exponent that we need on 4 to get 16 once we do the exponentiation. So, since,

we must have the following value of the logarithm.

(b)

This one is similar to the previous part. Let’s first convert to exponential form.

If you don’t know this answer right off the top of your head, start trying numbers. In other words, compute

,etc until you get 16. In this case we need an exponent of 4. Therefore, the value of this logarithm is,

Hopefully, you now have an idea on how to evaluate logarithms and are starting to get a grasp on the notation. There are a few more evaluations that we want to do however, we need to introduce some special logarithms that occur on a very regular basis. They are the common logarithm and the natural logarithm. Here are the definitions and notations that we will be using for these two logarithms.

So, the common logarithm is simply the log base 10, except we drop the “base 10” part of the notation. Similarly, the natural logarithm is simply the log base e with a different notation and where e is the same number that we saw in the previous section and is defined to be

Example 2 Evaluate each of the following logarithms.

(a)

(b)

Solution:

(a)

because

(b)

because

Properties of Logarithms

. This follows from the fact that .
. This follows from the fact that .
. This can be generalize out to .
. This can be generalize out to .

Properties 3 and 4 leads to a nice relationship between the logarithm and exponential function. Let’s first compute the following function compositions for

and

Recall from the section on inverse functions that this means that the exponential and logarithm functions are inverses of each other. This is a nice fact to remember on occasion.

We should also give the generalized version of Properties 3 and 4 in terms of both the natural and common logarithm as we’ll be seeing those in the next couple of sections on occasion.

Now, let’s take a look at some manipulation properties of the logarithm.

More Properties of Logarithms

For these properties we will assume that

and

If then .

We won’t be doing anything with the final property in this section; it is here only for the sake of completeness. We will be looking at this property in detail in a couple of sections.

The first two properties listed here can be a little confusing at first since on one side we’ve got a product or a quotient inside the logarithm and on the other side we’ve got a sum or difference of two logarithms. We will just need to be careful with these properties and make sure to use them correctly.

Also, note that there are no rules on how to break up the logarithm of the sum or difference of two terms. To be clear about this let’s note the following,

Be careful with these and do not try to use these as they simply aren’t true.

Note that all of the properties given to this point are valid for both the common and natural logarithms. We just didn’t write them out explicitly using the notation for these two logarithms, the properties do hold for them nonetheless

Now, let’s see some examples of how to use these properties.

Example 4 Simplify each of the following logarithms.

(a)

(b)

Solution :

(a)

Note that we can’t use Property 7 to bring the 3 and the 5 down into the front of the logarithm at this point. In order to use Property 7 the whole term in the logarithm needs to be raised to the power. In this case the two exponents are only on individual terms in the logarithm and so Property 7 can’t be used here.

We do, however, have a product inside the logarithm so we can use Property 5 on this logarithm.

Now that we’ve done this we can use Property 7 on each of these individual logarithms to get the final simplified answer.

(b)

In this case we’ve got a product and a quotient in the logarithm. In these cases it is almost always best to deal with the quotient before dealing with the product. Here is the first step in this part.