Properties of the Exponential functions

For x and y real numbers:

1. a^xa^y = a^{x + y}
   example: 2³2⁵ = 2⁸
   
   
2. (a^x)^y = a^xy
   example: (4²)⁵ = 4¹⁰
   
   
3. (ab)^x = a^xb^x
   example: (3*7)³ = 3³7³
   
   
4. (a/b)^x = a^x/b^x
   example: (3/5)³ = 3³/5³
   
   
5. a^x/a^y = a^{x - y}
   example: 5⁷/5⁴ = 5³
   
   

Example 1 : Simplify the following expression

Solution to Example 1:

1. Use property (1) above to write the term 2^{x + 1} as 2^x2 in the given expression
   2^x - 2^{x + 1} = 2^x - 2^x2
   
   
2. Factor 2^x out
   2^x - 2^{x + 1} = 2^x(1 - 2)
   
   
3. Simplify to obtain
   2^x - 2^{x + 1} = -2^x
   
   

Matched Exercise 1: Simplify the following expression

Example 2 : Find parameters A and k so that f(1) = 1 and f(2) = 2, where f is an exponential function given by

Solution to Example 2:

1. Use the fact that f(1) = 1 to obtain
   1 = Ae^k
   
   
2. Now use f(2) = 2 to obtain
   2 = Ae^2k
   
   
3. Multiply all terms of the equation obtained in step 1 by -2
   -2 = -2Ae^k
   
   
4. Add the equation in steps 2 and 3
   2 - 2 = Ae^2k - 2Ae^k
   
   
5. and simplify
   Ae^2k - 2Ae^k = 0
   
   
6. Factor Ae^k out.
   Ae^k(e^k - 2) = 0
   
   
7. Neither A nor e^k can be equal to zero. Therefore
   (e^k - 2) = 0
   
   
8. Rewrite the above equation as follows
   e^k = 2
   
   
9. Take the ln of both sides
   k = ln(2)
   
   
10. To obtain parameterA, substitute the value of k obtained in the equation obtained in step 1.
    1 = Ae^ln(2)
    
    
11. Simplify and solve for A.
    A = 1/2
    
    
12. Function f is given by
    f(x) = (1/2)e^xln(2)
    
    
13. Which can be written as
    f(x) = (1/2)(e^ln(2))^x
    
    
14. and simplified to
    f(x) = 2^{x - 1}
    
    

Check answer
f(1) = 2^{1 - 1}
= 1
f(2) = 2^{2 - 1}
= 2

Matched Exercise 2: Find parameters A and k so that f(1) = 3 and f(2) = 9, where f is an exponential function given by