Chapter 7 Exponential Functions PDF

CHAPTER 7: EXPONENTIAL FUNCTIONS This chapter introduces the important class of functions called exponential function. This function is used extensively in modeling and solving wide variety of real-world problems, including growth of money at compound interest, population growth, radioactive decay,...

CHAPTER 7: EXPONENTIAL FUNCTIONS This chapter introduces the important class of functions called exponential function. This function is used extensively in modeling and solving wide variety of real-world problems, including growth of money at compound interest, population growth, radioactive decay, etc. 7.1 Exponential Function 𝑓(𝑥) = 2𝑥 𝑓(𝑥) = 𝑥 2 𝑓(𝑥) = 2𝑥 𝑦 𝑦 𝑦 𝑥 𝑥 𝑥 Linear Function Quadratic Function Exponential Function An exponential function is a mathematical function of the following form: 𝑓(𝑥) = 𝑏 𝑥 for 𝑏 > 0, 𝑎 ≠ 1 where 𝑥 is a variable in the exponent (or index or power), and 𝑏 is a constant called base of the function. The domain of f is the set of all real numbers and the range of f is the set of all positive real numbers. 7.2 Base e Exponential Function The function defined by 𝑓(𝑥) = 𝑒 𝑥 is called the natural exponential function, where the base 𝑒 is an irrational number like 𝜋 and equal to approximately 2.718281828… 91 7.3 Laws and Properties of Exponents. 1. 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛 Example: 𝑥 2 ∙ 𝑥 3 = 𝑥 2+3 = 𝑥 5 𝑎𝑚 𝑥2 2. = 𝑎𝑚−𝑛 Example: 𝑥 3 = 𝑥 2−3 = 𝑥 −1 𝑎𝑛 3. (𝑎𝑚 )𝑛 = 𝑎𝑚∙𝑛 Example: (𝑥 2 )3 = 𝑥 2∙3 = 𝑥 6 4. (𝑎𝑏)𝑚 = 𝑎𝑚 ∙ 𝑏 𝑚 Example: (2𝑥)3 = 23 ∙ 𝑥 3 = 8𝑥 3 𝑎 𝑛 𝑎𝑛 𝑥 3 𝑥3 𝑥3 5. (𝑏) = 𝑏𝑛 , 𝑏 ≠ 0 Example: (2) = 23 = 8 1 1 1 6. 𝑎−𝑛 = 𝑎𝑛 Example: 2−2 = 22 = 4 1⁄ 𝑚 1⁄ 3 7. 𝑎 𝑚 = √𝑎 Example: 8 3 = √8 = 2 𝑛⁄ 𝑚 2⁄ 3 3 8. 𝑎 𝑚 = √𝑎 𝑛 Example: 8 3 = √82 = √64 = 4 9. 𝑎0 = 1 Example: 𝑥 0 = 1 ; 8𝑒 0 = 8 ∙ 1 = 8 10. 𝑎 𝑥 = 𝑎 𝑦 if and only if x = y. 11. 𝑎 𝑥 = 𝑏 𝑥 if and only if a = b and for x ≠ 0 Example 7.1: Simplify each of the following expressions. a) 2a 3  4a 2  6a 5 ( b) 9  (e −1.4 x ) 2  2e 2 x ) 5 92  Exponential Function Example 7.2: Solve the following exponential equations. a) (1 − x) 5 = (2 x − 1) 5 b) 2 2 x −1 = 16 1 9 2 x = 27  91− x c) 8x −1 − =0 d) 32 x e) 2 x = 5 Question: Can we rewrite both sides with same bases? If not, how can we solve this equation? 93  Exponential Function Tutorial 7 1. Simplify the following expressions. a. (a 2 ) 2n  (a 3−n ) 2 b. (a 3b 2 ) 3  (ab) 4 1 6 −12 6 c. (64 p q ) d. 2 n  21−2 n  2 3n+ 2 e. 5 4 n + 2  1251− n  25 2 n 2 1 f. (27 p 6 q −3 ) 3  (4 pqr ) 2  (16q − 2 r 2 ) 2 2. Solve the following exponential equations. a) 362 x+1 = 216 b) 3x = 4 35 1 c) 16 x + 2 = 4  2 x +1 x 25 d) 125x −1 = 5x +3 e) 48 x −4  2563 x −5 = 2569 x + 2 f) 2x = 162 x−3 2 Answers 1. a) a 6+ 2 n 1 2. a) 4 b) a 5 b 2 5 c) 2 pq −2 b) 4 d) 2 −4 n −1 c) − 1 e) 511n −1 d) 1 f) 36𝑝6 𝑞𝑟 e) −2 f) 2 and 6 94