Unit 11 – Exponential and Logarithmic Functions

Unit 11.1 Inverse Functions

We are going to look at exponential functions and logarithmic functions in this unit. For example is an exponential function and is a logarithmic function.

These functions relate to each other. They are inverses of each other. So let’s look at inverse functions first.

Review:

Define domain: all the inputs, the x values

Define range: all the outputs, the y values

Give an example of a function with an unlimited domain (i.e. all real numbers):

Give an example of a function with a limited domain. What is it’s limitation?: the number -2 is not in the domain as it makes the denominator zero and thus the function is undefined.

What is the range of the two example functions: the range of the first function is all real numbers and the range of the second is all real numbers with the exception of zero.

One-to-One functions: In a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to just one x-value.

What is the difference between this definition and the definition of a function we talked about earlier this year? A function only requires that each x-value have one and only one y-value. A y-value may “belong” to more than one x-value. This is not true in a one-to-one function.

Only one-to-one functions have an inverse.

Let’s define function

If we interchange the x and y values we get the inverse of function G.

Because we interchange the x and y values to get the inverse, it follows that the domain of the original function is the range of the inverse and the range of the original function becomes the domain of the inverse.

The function notation for inverse functions is . Notice that once again we are using notation that looks like something else. You’re just going to have to go by the context of the problem. In this context we are looking at an inverse function, not a negative exponent.

To find an inverse function you first have to determine if the function is a one-to-one function. If it isn’t, then it doesn’t have an inverse and you can stop looking!

Example:

This is not a one-to-one function because the y-value 2 corresponds to two x-values. No inverse.

is a on-to-one function and thus has an inverse

Horizontal Line Test:

We use a Vertical Line Test to help us determine if a particular equation is a function. In a similar fashion we use a Horizontal Line Test to help us determine if a particular equation is a one-to-one function. If every horizontal line intersects the graph of the function at only one point then the function is a one-to-one function. (Note the equation has to first pass the vertical line test and be a function.

Examples to put on board:

Finding the equation of the inverse of a function

  1. Rewrite the function notation as a “y =” form of equation.
  2. Interchange the y and the x.
  3. Solve for y.
  4. Rewrite in inverse function notation by replacing the y with

Example: Use the function from above .

  1. Rewrite as “y =”
  2. Interchange the y and the x
  3. Solve for y
  4. Rewrite in inverse function notation

Examples:

  1. inverse is
  2. inverse is
  3. there is no inverse this is not a one-to-one function. Why not?

An inverse function “undoes” it’s related function. One way to look at this is you will end up back where you started. For example, plug a number into the function in #1 above, take the output and plug it into the inverse. What do you get?

Unit 11.2 – Exponential Function

For a > 0 and a  1 and all real numbers x, defines the exponential function with base a.

Graph (blue line) and (green line)

This graph is typical of exponential functions where a > 1. The larger the value of a the faster the graph rises.

Graph (blue line) and (green line)

Here as x gets larger y gets smaller.

Characteristics of the graph of

  1. The graph contains the point (0, 1).
  2. If a > 1 the graph rises from left to right. If 0 < a < 1, the graph falls from left to right. Both go from 2nd to 1st quadrants.
  3. The x-axis is an asymptote.
  4. The domain is and the range is .

What happens when you graph a more complicated exponential function such as ? (Very similar to except it is shifted to the right and rises more rapidly.)

How would we solve ?

Property for solving an exponential Equation

For a > 0 and a  1, if .

Solving an Exponential Equation

  1. Each side must have the same base. If the two sides do not have the same base, express each as a power of the same base.
  2. Simplify exponents, if necessary, using the rules of exponents.
  3. Set exponents equal using the exponential equation property.
  4. Solve the equation from Step 3.

(Note: for equations such as we need to use a different method because Step 1 is difficult to do.)

Example:

Examples:

Unit 11.3 Logarithmic Functions

What is a logarithm?

A logarithm is an exponent. Remember in the previous unit I said that we couldn’t solve because it is difficult to express both sides of the equation as expressions with the same base? Well, there are ways to use logarithms to solve equations such as this. (We’ve got a ways to go before we’re ready for that. (check page 695)).

Logarithm

For all positive numbers a, with a  1, and all positive numbers x, . This is saying the expression represents the exponent to which the base a must be raised in order to get x.

Solving logarithmic equations

For any positive real number b, with b  1,

Unit 11.4 Properties of Logarithms

Product Rule for Logarithms

The logarithm of a product is equal to the sum of the logarithms of the factors.

Proof of the Product Rule

Examples:

Quotient Rule for Logarithms

If x, y, and b are positive real numbers, where b  1, then

The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.

Examples:

? any restrictions on x?

this illustrates an important point. When solving logs, if you can convert a log to a number then do so.

Power Rule for Logarithms

If x and b are positive real numbers, where b  1, and if r is any real number, then

Recall this example from the Product Rule:

Examples:

when using the power rule with logarithms of expressions involving radicals, first rewrite the radical expression with a rational exponent.

Special Properties of Logarithms

If b > 0 and b  1, then

and

The second is the easiest to understand.

To prove the first property:

Let

Properties of Logarithms

Product Rule

Quotient Rule

Power Rule

Special Properties and

Writing Logarithms in Alternative Forms

Unit 11 - Student NotesPage 1 of 8