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

• 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
  
• Factor 2^x out
  2^x - 2^{x + 1} = 2^x(1 - 2)
  
• Simplify to obtain
  2^x - 2^{x + 1} = -2^x

Matched Exercise 1

Example 2

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

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