Tutorial on Exponential Functions (1)



This is a tutorial on exponential functions to further understand the properties of the these functions. Examples with detailed solutions and explanations are included.

Properties of the Exponential functions


For x and y real numbers:

  1. axay = ax + y
    example: 2325 = 28

  2. (ax)y = axy
    example: (42)5 = 410

  3. (ab)x = axbx
    example: (3*7)3 = 3373

  4. (a/b)x = ax/bx
    example: (3/5)3 = 33/53

  5. ax/ay = ax - y
    example: 57/54 = 53


Example 1 : Simplify the following expression

2x - 2x + 1

Solution to Example 1:

  1. Use property (1) above to write the term 2x + 1 as 2x2 in the given expression
    2x - 2x + 1 = 2x - 2x2

  2. Factor 2x out
    2x - 2x + 1 = 2x(1 - 2)

  3. Simplify to obtain
    2x - 2x + 1 = -2x

Matched Exercise 1: Simplify the following expression

3x - 3x + 1


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

f(x) = Aekx

Solution to Example 2:

  1. Use the fact that f(1) = 1 to obtain
    1 = Aek

  2. Now use f(2) = 2 to obtain
    2 = Ae2k

  3. Multiply all terms of the equation obtained in step 1 by -2
    -2 = -2Aek

  4. Add the equation in steps 2 and 3
    2 - 2 = Ae2k - 2Aek

  5. and simplify
    Ae2k - 2Aek = 0

  6. Factor Aek out.
    Aek(ek - 2) = 0

  7. Neither A nor ek can be equal to zero. Therefore
    (ek - 2) = 0

  8. Rewrite the above equation as follows
    ek = 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 = Aeln(2)

  11. Simplify and solve for A.
    A = 1/2

  12. Function f is given by
    f(x) = (1/2)exln(2)

  13. Which can be written as
    f(x) = (1/2)(eln(2))x

  14. and simplified to
    f(x) = 2x - 1

Check answer
f(1) = 21 - 1
= 1
f(2) = 22 - 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

f(x) = Aekx


More references and links related to the exponential functions in this web site.


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Updated: 2 April 2013

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