1. WARM UP 1. Find the equation of the inverse relation for 2. Find for f(x) = 8x – 1. Switch y and x in y = 8x – 1 x = 8y – 1 x + 1 = 8y So

2. EXPONENTIAL & LOGARITHMIC FUNCTIONS

3.  Graph exponential functions OBJECTIVES  Graph logarithmic functions  Model real-world problems that involve exponential and logarithmic functions.

4. 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:  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  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.

5. EXPONENTIAL FUNCTIONS  Exponential functions are defined using exponential notation. Definition The function, where a is some positive real-number constant different from 1, is called the exponential function, base a.  Here are some exponential functions:  Note that the variable is the exponent. The following are not exponential functions:  Note that the variable is not the exponent.

6. EXAMPLE 1 Graph. Use the graph to approximate, We find some solutions, plot them and then draw the graph. xy 01 12 24 38 ½ 2¼ -3 ⅛ Note that as x increases, the function values increase. Check this on a calculator. As x decreases, the function values decrease toward 0. To approximate we locate on the x-axis, at about 1.4. Then we find the corresponding function value. It is about 2.7.

7. TRY THIS… Graph. Use the graph to approximate. xy

8. 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.

9. TRY THIS… Graph. xy

10. EXAMPLE 3 Graph 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.

11. TRY THIS… Graph. xy

12. LOGARITHMIC FUNCTIONS Definition A 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. means

13. 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.

14. EXAMPLE 4 Graph The equation is equivalent to. The graph of is a xy 01 13 29 ⅓ -2 1/9 Since a = 1 for any a ≠ 0, the graph of for a has the x-intercept (1, 0) reflection of across the line y = x. We make a table of values for and then interchange x and y For : xy 10 31 92 1/3 1/9-2 For or :

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

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