WARM UP

1. Find the equation of the inverse relation for

2. Find for f(x) = 8x – 1.

xy−y3 =2 xy−x3 =2f −1(x)

Switch y and x in y = 8x – 1 x = 8y – 1

x + 1 = 8yx +18

=y

So f −1 =x+18

EXPONENTIAL & LOGARITHMIC FUNCTIONS

Graph exponential functions

OBJECTIVES

Graph logarithmic functions

Model real-world problems that involve exponential and logarithmic functions.

INTRODUCTION We have defined exponential notation for rational exponents.

Let us consider 2 . The number π has an unending decimal representation.

Now consider this sequence of numbers.3.1415926535…..

3, 3.1, 3.14, 3.141 3.1415 3.14159….. Each of these numbers is an approximation to . The

more decimal places, the better the approximation. Let us use these rational numbers to form a sequence as follows:

π

π

23 23.1 23.14 23.141 23.1415 23.14159 Each of the numbers in this sequence is already defined,

the exponent being rational. The numbers in this sequence get closer and closer to some real number. We define that number to be 2

π

We can define exponential notation for any irrational exponent in a similar way. Thus any exponential expressions , now has meaning, whether the exponent is rational or irrational.

ax ,a > 0

EXPONENTIAL FUNCTIONS Exponential functions are defined using

exponential notation.DefinitionThe function , where a is some positive real-

number constant different from 1, is called the exponential function, base a.

f (x)=ax

Here are some exponential functions:f (x)=2x g(x)=(

12)x h(x)=(0.178)x

Note that the variable is the exponent. The following are not exponential functions:f (x)=x2 g(x)=x

13 h(x)=x0.178

Note that the variable is not the exponent.

EXAMPLE 1Graph . Use the graph to approximate , y =2x 2 2

We find some solutions, plot them and then draw the graph.x y0 11 22 43 8-1 ½2 ¼-3 ⅛

Note that as x increases, the function values increase. Check this on a calculator.

2 2 ≈2.7

2

As x decreases, the function values decrease toward 0.To approximate we locate on the x-axis, at about 1.4.

22 2

Then we find the corresponding function value. It is about 2.7.

TRY THIS…Graph . Use the graph to approximate .

y =3x 312

x y

EXAMPLE 2 We can make comparisons between functions

using transformations.Graph We note that . Compare this with graphed in Example 1.Notice that the graph of approaches the y-axis more rapidly than the graph of The graph of is a shrinking of the graph of Knowing this allows us to graph at once. Each point on the graph of is moved half the distance to the y-axis.

y =4 x

4 x =(22 )x =22x

y =2x

y =22x

y =2x

y =22x

y =2x

y =2x

y =2xy =22xor4x

y =22x

TRY THIS…Graph .

x y

y =8x

EXAMPLE 3Graph

We could plot some points and connect them, but again let us note that or

Compare this with the graph of in Example 1. The graph of is a reflection, across the y-axis, of the graph of

Knowing this allows us to graph at once.

y =2x

y =2x

y =2x

y =12

x

(12)x =

12x =2−x

y =2−x

y =2−x

y =2−x 12x

TRY THIS…Graph .

x y

y =(13)x

LOGARITHMIC FUNCTIONSDefinitionA logarithmic function is the inverse of an

exponential function. One way to describe a logarithmic function is to

interchange variables in the equation y = a . Thus the following equation is logarithmic

For logarithmic functions we use the notation or log x which is read “log, base a, of x.”

Thus a logarithm is an exponent. That is, we use the symbol to denote the second coordinate of a function.

x

x =ay

loga (x) a

y =loga x means x =ay

loga xx =ay

MORE LOGARITHMIC FUNCTIONS

The most useful and interesting logarithmic functions are those for which a > 1. The graph of such a function is a reflection of across the line y = x. The domain of a logarithmic function is the st of all positive real numbers.

y =ax y =ax

y =ay

EXAMPLE 4Graph

The equation is equivalent to . The graph of is a

x y0 11 32 9-1 ⅓-2 1/

9

Since a = 1 for any a ≠ 0, the graph of for a has the x-intercept (1, 0)

y =log3 xy =log3 x

reflection of across the line y = x. We make a table of values for and then interchange x and y

x =3y x =3y

y =3x

y =3x

For : y =3x

x y1 03 19 2

1/3 -11/9 -2

For

or : x =3y

y =log3 x y =3x

y =log3 x

y =loga x

TRY THIS…Graph . What is the domain of this function? What is the range?

x y

y =log2 x

CH. 12.1 & 12.2Textbook pg. 519 #2, 6, 12 & 14 pg. 525 #2, 6, 14, 30 & 32