To Solve Equations which Involve Exponents. If b > 0 and b 1, and m and n are real numbers, then bn = bm if and only if n = m. Property 1. If the bases are equal, then the exponents must be equal. If the exponents are equal then the bases must be equal. Solution: Since the bases are equal, then the exponents are equal. Answer: Your Turn Problem #1

Solution: If the bases do not match, then use exponents to make the bases match. Rewrite one or both sides of the equation using a common base. Then use the property 1. As with all equations, a check can be performed to verify the answer. It is not always possible to make the bases match. Another method will be covered in a later section to solve these types of problems. 81 can be rewritten with a base of 3. Answer: Your Turn Problem #2

Solution: Your Turn Problem #3 Solution: Answer: Your Turn Problem #4

Solution: It is necessary to change both sides of the equation to match the bases. Answer: Your Turn Problem #5 Solution: It is necessary to change both sides of the equation to match the bases. Answer: Your Turn Problem #6

Solution: There are two bases on the left hand side. The first goal is to condense it so there is only one base. Your Turn Problem #7 Answer: Graphing Exponential Functions If b > 0 and b 1, then the function f defined by f(x) =bx where x is any real number, is called the exponential function with base b. Definition 1. A basic technique for graphing a function is to plot enough points to obtain an accurate shape of the graph.

Solution: Choose enough points to graph the function accurately. The domain is any real numbers, so choose positive and negative numbers for x. x y 0 1 1 2 2 4 -1 1/2 -2 1/4 -3 1/8 Next Slide It is important to choose the values for x which will give an accurate drawing of the graph. Choose the value for x which will give zero in the exponent. Choose values for x which will give positive and negative values in the exponent. y axis x axis (-2,1/4) (2,4) (0,1) (-1,1/2) (1,2)

Your Turn Problem #8 Answer: y axis x axis (2,9) (1,3) (-1,1/3) (0,1) (-2,1/9) (2,9) (0,1) (-1,1/3) (1,3)

Solution: Choose the value for x which will give zero in the exponent. Then choose at least two values for x to the left of that number and at least two values to the right of that number. x y 1 1 2 2 3 4 0 1/2 -1 1/4 y axis x axis (-1,1/4) (3,4) (1,1) (0,1/2) (2,2) Next Slide Again notice the graph does not go below the x axis. The x axis is then an asymptote for this function.

Your Turn Problem #9 Answer: y axis x axis (-2,9) (-3,3) (-5,1/3) (-6,1/9) (-2,9) (-4,1) (-5,1/3) (-3,3)

Note: Exponential functions of the form Solution: Choose the value for x which will give zero in the exponent. Then choose at least two values for x to the left of that number and at least two values to the right of that number. x y 0 -2 1 -2 ½ 2 -2 ¾ -1 -1 -2 1 Again notice the graph does not go below the line y = -3. The line y = -3 is then an asymptote for this function. y axis x axis (1, -2½ ) (-2,1) (0,-2) (-1,-1) (-2, -2¾) Next Slide Note: Exponential functions of the form f(x) = ax + k always have an asymptote at y = k.

Your Turn Problem #10 Answer: y axis x axis (-1, 5) (-2, 7) (1, 3½) (2, 3¼) (0,4) The homework may consist of other exponential functions whose graph is unlike the previous examples. The key is to choose enough points to accurately draw the graph. How may points is enough? Five is usually enough, however there may be times when five is not enough. The End B.R. 1-4-07