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:
  • Use property (1) above to write the term 2x + 1 as 2x2 in the given expression
    2x - 2x + 1 = 2x - 2x2
  • Factor 2x out
    2x - 2x + 1 = 2x(1 - 2)
  • 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) = A ek x

Solution to Example 2:
  • Use the fact that f(1) = 1 to obtain
    1 = Aek
  • Now use f(2) = 2 to obtain
    2 = A e2k
  • Multiply all terms of the equation obtained in step 1 by -2
    -2 = -2 A ek
  • Add the equation in steps 2 and 3
    2 - 2 = A e2 k - 2 A ek
  • and simplify
    A e2 k - 2 A ek = 0
  • Factor A ek out.
    A ek(ek - 2) = 0
  • Neither A nor ek can be equal to zero. Therefore
    (e k - 2) = 0
  • Rewrite the above equation as follows
    ek = 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 eln(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) = 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) = A e k x

More References and Links Related to Exponential Functions

Exponential Functions.
Exponential and Logarithmic Functions.
Solve Exponential and Logarithmic Equations (self test).