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Exponential Functions Questions with Solutions

Questions on exponential functions are presented along with their their detailed solutions and explanations.

Properties of the Exponential functions

For x and y real numbers:

  1. a x a y = a x + y
    example: 2 3 25 = 2 8
  2. (a x) y = a x y
    example: (4 2) 5 = 4 10
  3. (a b) x = a x b x
    example: (3 7)3 = 33 73
  4. (a / b)x = a x / b x
    example: (3 / 5)3 = 3 3 / 5 3
  5. a x / a y = a x - y
    example: 5 7 / 5 4 = 5 3

Questions with Detailed Solutions and Explanations

Question 1
Simplify the following expression
2 x - 2 x + 1
Solution to Question 1
  • Use property (1) above to write the term 2 x + 1 as 2 x 2 in the given expression
    2 x - 2 x + 1 = 2 x - 2 x 2
  • Factor 2x out
    2 x - 2 x + 1 = 2 x(1 - 2)
  • Simplify to obtain
    2 x - 2 x + 1 = - 2 x

Question 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 e k x

Solution to Question 2
  • Use the fact that f(1) = 1 to obtain
    1 = A e k
  • Now use f(2) = 2 to obtain
    2 = A e 2 k
  • Rewrite the above equation as
    2 = A ek ek
  • Use the first equation 1 = A e k obtained in the first step to rewrite 2 = A ek ek as
    2 = e k
  • Take the ln of both sides and simplify to obtain
    k = ln (2)
  • To obtain parameter A, substitute the value of k obtained in the equation 1 = A e k.
    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 against given information
f(1) = 2 1 - 1
= 1
f(2) = 2 2 - 1
= 2

Question 3
The populations of 2 cities grow according to the exponential functions

P1(t) = 100 e 0.013 t
P2(t) = 110 e 0.008 t

where P1 and P2 are the populations (in thousands) of cities A and B respectively; t is the time in years such that t is positive and t = 0 corresponds to the year 2004.
When will the populations of the two cities be equal and what will be their populations?

Solution to Question 3

  • Let t = t' be the time when P1 and P2 are equal, this leads to the following equation in t'
    100 e 0.013 t' = 110 e 0.008 t'
  • Divide both side of the above equation by 100 e 0.008 t' and simplify to obtain
    e 0.013 t' / e 0.008 t' = 110/100
  • Use property of exponential functions a x / ay = a x - y and simplify 110/100 to rewrite the above equation as follows
    e 0.013 t'- 0.008 t' = 1.1
  • Simplify the exponent in the left side
    e 0.005 t' = 1.1
  • Rewrite the above in logarithmic form (or take the ln of both sides) to obtain
    0.005 t' = ln 1.1
  • Solve for t' and round the answer to the nearest unit.
    t' = (ln 1.1) / 0.005.
    t' is approximately equal to 19 years.
    the year will be: 2004 + 19 = 2023.
  • Find the populations when t = t' = 19 years. Use any of the function P1 or P2 since they are equal at t = t'
    P1(t') = 100 e 0.013*19
    P1(t') is approximately equal to 128 thousands.
    For checking, the graphical solution to the above problem is shown below.

    graphical solution of the above question.

Question 4
The amount A of a radioactive substance decays according to the exponential function

A(t) = A 0 e r t

where A0 is the initial amount (at t = 0) and t is the time in days (t ≥ 0). Find r, to three decimal places, if the half life of this radioactive substance is 10 days.

Solution to Question 4

  • At t = 10 days, the amount A of the substance would be equal to half the initial amount A0 (definition of half life)
    A 0 e r10 = A 0 / 2
  • Divide both side of the above equation by A0
    e 10 r = 1 / 2
  • Rewrite the above equation in logarithmic form (or take ln of both sides) to obtain
    10 r = ln (1/2)
  • Solve for r
    r = 0.1 ln(1/2)
  • Approximate r to 3 decimal places.
    r is approximately equal to -0.069.
    For checking, the graph of A(t) = 100 e-0.069t is shown below.
    Note at t = 0 A = 100 (initial amount) and at t = 10 (half life), A is approximately equal to 50 which half the initial amount 100.
    graphical solution to the above question.

More Questions with Answers

  1. Simplify the following expression
    3 x + 2 3 x + 2 3 x + 1
  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
  3. The populations of 2 cities grow according to the exponential functions
    P1(t) = 120 e 0.011 t
    P2(t) = 125 e 0.007 t

    where P1 and P2 are the populations (in thousands) of cities A and B respectively. t is the time in years such that t is positive and t = 0 corresponds to the year 2004.
    When were the populations of the two cities equal and what were their populations?
  4. The amount A of a radioactive substance decays according to the exponential function
    A(t) = A 0 e r t

    where A0 is the initial amount (at t = 0) and t is the time in days (t ≥ 0). Find r, to three decimal places, if the the half life of this radioactive substance is 20 days.

    Answers to the Above Questions


    1. 3 x + 2 3 x + 2 3 x + 1
      = 3 x + 2 3 x + 2 3 x 3
      = 3 x + 2 3 x + 6 3 x
      = 3 x(1 + 2 + 6)
      = 9 3 x = 3 2 3 x = 3 x + 2

    2. f(1) = A e k = 3 and f(2) = A e 2 k = 9
      A e 2 k = 9 can be written as
      A e k e k = 9 and we also know that A e k = 3; hence
      3 e k = 9
      which simplifies to
      e k = 3
      Take ln of both sides to solve for k and obtain
      k = ln(3)
      Substitute k by ln(3) in the equation A e k = 3 and simplify to obtain
      A = 1.

    3. Find t such that
      120 e 0.011 t = 125 e 0.007 t
      e 0.011 t - 0.007 t = 125 / 120
      Simplify and take ln of both sides
      0.004 t = ln (125 / 120)
      t = 10.2 years
      The two populations were equal in 2004 + 10 = 2014
      The population of each city in 2014 was
      120 e 0.011 10 = 134 thousands.

    4. Solve for r the equation
      A 0 e 20 r = A 0 / 2
      Simplify
      e 20 r = 1/2
      Take ln of both sides
      20 r = ln(1/2)
      r = - 0.035

    More References and Links Related to Exponential Functions

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

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