Building Exponential Functions

1
Building Exponential Functions

• A Miscellany of Features of Logarithmic and
  Exponential Functions

2
Population Growth / Food Production

• A pair of students will model an exponential
  function and a linear function separately but
  simultaneously.
• Population Food
• 10,000,000 15,000,000
• 10,000,000 15,000,000
• ANS1.02 ANS 500,000
• What do the specific numbers represent?

3
Population Growth / Food Production

• The food production begins with the ability to
  feed more than the population. Does that
  production continue to be able to stay ahead of
  the population growth?
• What is the population function? The food
  function?
• P(t) F(t)

10,000,0001.02t
15,000,000500,000t
4
Population Growth / Food Production

• Graph P(t) and F(t) on your calculator.
• Describe the results.
• What conclusion can be made about
  exponential functions and linear
  functions together?
• Double the initial amount of food and simulate
  again.
• Triple the rate at which food is produced and
  simulate again.
• Conclusions?

5
Building Exponential Functions

• Given that a generic exponential function is
• y abx
• Suppose that the exponential function passes
  through the two points (0 , 3) and (1 , 6).
• y abx ? 3 a b0 ? a 3
• y 3bx ? 6 3b1 ? b 2 ? y 32x

6
Building Exponential Functions

• Build the exponential function which passes
  through (0 , 7) and (2 , 63)

7
Building Exponential Functions

• Build the exponential function which passes
  through (0 , 7) and (4 , 104)

8
Building Exponential Functions

• Build the exponential function which passes
  through (2 , 7) and (4 , 28)

9
Building Exponential Functions

• Build the exponential function which passes
  through (3 , 1) and (8 , 209)

10
Matching Graphs to Functions
Match each function with a graph above
f(x) 25x g(x) 9 5x h(x) 2 12x
j(x) 2 (0.5)x
11
What is Concavity?
y 3 4x Find the rate of change from (0 ,
) to (1 , ). Find the rate of change from (6
, ) to (7, ). Compare the rates at lower
xs to higher xs.
12
What is Concavity?
y 10 0.2x Find the rate of change from (0 ,
) to (1 , ). Find the rate of change from
(10 , ) to (11, ). Compare the rates at
lower xs to higher xs.
13
What is Concavity?
y log(x) Find the rate of change from (0.5 ,
) to (1 , ). Find the rate of change from (4
, ) to (4.5, ). Compare the rates at
lower xs to higher xs.
14
Solving Harder Exponential Equations

• Solve 6 5x 73
• 5x 12.16666
• x log 5 log (12.16666)
• x 1.553

15
Solving Harder Exponential Equations

• Solve 8 9x 4 20x
• 1) You can take the log of both sides
  immediately. .. Or
• 2) You can reduce one of the multipliers
  before taking logs.

16
Solving Harder Exponential Equations

• Solve 8 9x 4 20x
• log (8 9x ) log (4 20x)
• log 8 x log 9 log 4 x log 20
• x(log 9 log 20) log 4 log 8
• x log(4/8) / log (9/20) 0.868

17
Solving Harder Exponential Equations

• Solve 11 6x 20 14x
• There are no solutions. Why?

18
Solving Logarithmic Equations

• Solve ln (x 2) ln (2x 3) 2 ln x
• ln (x-2) (2x - 3) ln x2
• (x 2) (2x 3) x2
• 2x2 7x 6 x2
• x2 7x 6 0
• (x 6) (x 1) 0
• x 6 x 1 ? only x
  6 is in the domain of the log
  function.

19
Solving Logarithmic Equations

• Solve log (x) log (2x 1) 0
• log (x / 2x 1) 0
• 100 x / 2x 1
• 1 x / 2x 1
• 2x 1 x ( cross multiply)
• x 1