Exponential and Logarithmic Functions
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
I.EXPONENTIAL FUNCTION A. Definition An exponential function is a function defined by f(x) = ax , where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers.
B. Evaluating Exponential Functions
1. Given: f(x) = 2x, find a. f(3) = ____ b. f(5) = _____ c. f(-2) = ______ d. f(-4) = ______
2. Evaluate f(x) = ( 1)x if 2 a. x = 2 ____ b. x = 4 _____ c. x = -3 ______ d. x = -4 _______
C. Graphing Exponential Functions On the same Cartesian Coordinate plane, sketch the graphs of each set of exponential function
1. a) f(x) = 2x b. f(x) = 3x c. f(x) = 4x
2. a) f(x) = 2x b. f(x) = 2x + 1 c. f(x) = 2x – 1
3. a) f(x) = 2x b. f(x) = 2x + 2 c. f(x) = 2x – 3
4. a) g(x) = 2- x b. g(x) = 3- x c. g(x) = 4 – x
D. The Property of Equality for Exponential Equations Let a, b and c be real numbers and a≠ 0, then ab = ac if and only if b = c
Examples:
1. 32x = 36 2. 23x = 8 3. 643x = 8 4. 10- x = 1/10000 5. 43x = 16x + 2 6. 16- x = 1/64 7. 93x = [ 1/3]5 8.5x+2 – 5x + 1 + 5x = 2625
Do as directed: A. Evaluate: 1. If f(x) = 3x, what is a. f( 3)?= ____ b. f( 4) = _____ c. f( -2) = _____ d. f( -4) = _____ 2. What is g(x) = [ 1/3 ]x if a.x = 2 ____ b. x = 4 ______ c. x = -3 ______ d. x = -4 ______
B. Solve for x 3. 2x = 128 6. 243x = 3 4. 3x = 81 7. 2x + 2 + 2x + 1 + 2x = 896 5. 42x = 8x + 1 8. 272x – 2 = 95 – x
C. Challenge!!! 9. If x is real and x64 = 64, what is x32? 10. Find the value of xy if 2x = 7 and 7y = 64. 11. If 183 = 2x •3y, find the integer values of x and y. 12. There are about 1,000 bacteria in a certain culture. If the amount doubles every 2
Hours, about how many bacteria would there be after 8 hours? 13. A radioactive substance is decaying (it is changing into
I.EXPONENTIAL FUNCTION A. Definition An exponential function is a function defined by f(x) = ax , where a > 0 and a ≠ 1. The domain of the function is the set of real numbers and the range is the set of positive numbers.
B. Evaluating Exponential Functions
1. Given: f(x) = 2x, find a. f(3) = ____ b. f(5) = _____ c. f(-2) = ______ d. f(-4) = ______
2. Evaluate f(x) = ( 1)x if 2 a. x = 2 ____ b. x = 4 _____ c. x = -3 ______ d. x = -4 _______
C. Graphing Exponential Functions On the same Cartesian Coordinate plane, sketch the graphs of each set of exponential function
1. a) f(x) = 2x b. f(x) = 3x c. f(x) = 4x
2. a) f(x) = 2x b. f(x) = 2x + 1 c. f(x) = 2x – 1
3. a) f(x) = 2x b. f(x) = 2x + 2 c. f(x) = 2x – 3
4. a) g(x) = 2- x b. g(x) = 3- x c. g(x) = 4 – x
D. The Property of Equality for Exponential Equations Let a, b and c be real numbers and a≠ 0, then ab = ac if and only if b = c
Examples:
1. 32x = 36 2. 23x = 8 3. 643x = 8 4. 10- x = 1/10000 5. 43x = 16x + 2 6. 16- x = 1/64 7. 93x = [ 1/3]5 8.5x+2 – 5x + 1 + 5x = 2625
Do as directed: A. Evaluate: 1. If f(x) = 3x, what is a. f( 3)?= ____ b. f( 4) = _____ c. f( -2) = _____ d. f( -4) = _____ 2. What is g(x) = [ 1/3 ]x if a.x = 2 ____ b. x = 4 ______ c. x = -3 ______ d. x = -4 ______
B. Solve for x 3. 2x = 128 6. 243x = 3 4. 3x = 81 7. 2x + 2 + 2x + 1 + 2x = 896 5. 42x = 8x + 1 8. 272x – 2 = 95 – x
C. Challenge!!! 9. If x is real and x64 = 64, what is x32? 10. Find the value of xy if 2x = 7 and 7y = 64. 11. If 183 = 2x •3y, find the integer values of x and y. 12. There are about 1,000 bacteria in a certain culture. If the amount doubles every 2
Hours, about how many bacteria would there be after 8 hours? 13. A radioactive substance is decaying (it is changing into