Building Exponential
Functions
A Miscellany of Features of
Logarithmic and Exponential
Functions
Population Growth / Food
Production
A pair of students will model an exponential function
and a linear function separately but simultaneously.
Population:
10,000,000
Food:
15,000,000
10,000,000
ANS*1.02
15,000,000
ANS + 500,000
What do the specific numbers represent?
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) = 10,000,000*1.02t
F(t) = 15,000,000+500,000*t
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?
Building Exponential Functions
Given that a generic exponential function is
y = a*bx
Suppose that the exponential function passes
through the two points (0 , 3) and (1 , 6).
y = a*bx  3 = a * b0  a = 3
y = 3bx  6 = 3b1  b= 2  y = 3*2x
Building Exponential Functions
Build the exponential function which passes through
(0 , 7) and (2 , 63)
Building Exponential Functions
Build the exponential function which passes through
(0 , 7) and (4 , 104)
Building Exponential Functions
Build the exponential function which passes through
(2 , 7) and (4 , 28)
Building Exponential Functions
Build the exponential function which passes through
(3 , 1) and (8 , 209)
Matching Graphs to Functions
Match each function with a graph above:
f(x) = 2*5x g(x) = 9 * 5x h(x) = 2 * 12x
j(x) = 2 * (0.5)x
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 x’s to
higher x’s.
What is Concavity?
y = 10 * 0.2x
Find the rate of
change from (0 ,
(1 , ).
) to
Find the rate of
change from (10 , )
to (11,
).
Compare the rates at
lower x’s to higher
x’s.
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 x’s to higher x’s.
Solving Harder Exponential
Equations
Solve:
6 * 5x = 73
5x = 12.16666
x log 5 = log (12.16666)
x = 1.553
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.
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
Solving Harder Exponential
Equations
Solve: 11 * 6x = 20 * 14x
There are no solutions. Why?
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.
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