Name_________________________________________________
Pre-Calculus
3.4 Exponential and Log Equations
3.5 Modeling Exponential and Log Equations
Date____________________________
Period_____________________
CTM, Does this Make Sense, and True/False
In every exercise below a mistake was made. Please find the mistake, and explain how to solve it correctly.
1.
Solve the equation
4e x  9
Incorrect Solution:
Correct Solution:
ln  4e x   ln 9
Take Natural Log of both sides
4 x  ln 9
Apply Properties of Logs
x
Solve for x
ln 9
 0.55
4
Isolate exponential part
Take Natural Log of both sides
Apply properties of Logs
Solve the equation
log x  log3  1
Incorrect Solution:
Correct Solution:
Apply the Product Property
11  3x
Change to exponents
x
Solve for x
3.
Solve the equation
log  3x   1
1
3
Correct
101  3x
x
10
 3.33
3
log x  log  x  3  1
Incorrect Solution:
Correct Solution:
log  x 2  3x   1
Correct
101  x2  3x
Correct
0   x  5 x  2 
Correct
Factor
Solve
x  5, x  2
x  5, x  2
Apply the Product Property
Change to exponents
9
4
9
ln  e x   ln  
4
9
x  ln  
4
9
x  ln    0.8109
4
Solve
2.
ex 
4.
Solve the equation
log x  log 2  log5
Incorrect Solution:
Correct Solution:
Apply the Product Property
log  x  2   log 5
Apply the Product Property
log  2 x   log 5
One-to-One Log can cancel
x25
One-to-One Log can cancel
2x  5
Solve for x.
x
Solve for x.
5.
x3
5
2
The city of Orlando, Florida, has a population that is growing at 7% a year, compounding continuously. If there were 1.1 million
people in the greater Orlando area in 2006, approximately how many people will be there in 2016? Use the formula
to create an equation and then use it to solve.
Incorrect Solution:
Use the growth model
Correct Solution:
A  Pert
Let P  1.1, r  7, t  10
A  1.1e
Approximate
A  2.8  1030
6.
A  Pert
7 10
Use the growth model
Let
P  1.1, r  0.07, t  10
Approximate
A  Pert
A  1.1e
0.07 10
A  2.215
The city of San Antonio, Texas, has a population that is growing at 5% a year, compounding continuously. If there were 1.3
million people in the greater San Antonio area in 2006, approximately how many people will be there in 2016? Use the formula
A  Pert to create an equation and then use it to solve.
Incorrect Solution:
Use the growth model
Let P  1.3, r  5, t  10
Approximate
Correct Solution:
A  Pert
A  1.3e
510
A  26.7  1021
Use the growth model
Let
P  1.3, r  0.05, t  10
Approximate
A  Pert
A  1.3e
0.0510 
A  2.1433
In every exercises below, determine whether each statement makes sense or does not make sense. Explain
why you think this way.
7. Because the equations 2 x  15 and 2 x  16 are similar, I MUST solve them using the same method.
Does not make sense.
different ways.
8.
Because the equations
Does not make sense.
where
2x  16
2x  24 , where 2x  15 cannot be changed to 2 to a power, thus can be solved in
log  3x  1  5 and log  3x  1  log 5 are similar, I MUST solve them using the same method
log  3x  1  5
log  3x  1  log 5
3x  1  5
105  3x  1
3x  4
100,000  3x  1
x
99,999  3x
x  33,333 ,
4
, therefore different methods used and different
3
answers obtained.
9.
I can solve
2x  15 by writing the equation in logarithmic form.
Makes sense.
10. It is important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive
numbers in the original equation.
Makes sense.
11. I used an exponential model with a POSTIVE growth rate to describe the depreciation in my car’s value over four years.
Does not make sense. Should be NEGATIVE growth rate.
12. After 100 years, a population whose growth rate is 3% will have three times as many people as a population growth rate is 1%.
Does not make sense. Need to know starting populations in order to then compare population after growth period.
13. When I used an exponential function to model Russia’s declining population, the growth rate k was negative.
Makes sense.
14. Because carbon-14 decays exponentially, carbon dating can determine the ages of ancient fossils.
Makes sense
Determine whether each statement below is TRUE or FALSE. If the statement is false, make the necessary
changes to produce a TRUE statement.
15. log  x  3  2 , therefore, e2  x  3
False. 102  x  3
100  x  3
97  x
4
16. log  7 x  3  log  2 x  5  4 , therefore, 10   7 x  3   2 x  5
False.
7x  3
4
2x  5
7x  3
104 
2x  5
7x  3
10000 
2x  5
20000 x  50000  7 x  3
19993 x  49997
x  2.5007
log
17. Examples of exponential equations include: 10x  5.71
False x10  5.71 is NOT exponential
18.
e x  2 has no solution.
True because x  ln  2 
19. Using the equations,
ex  0.72
x10  5.71
x
A  33.1e0.009t for Canada and A  28.2e0.034t for Uganda, where A is measured in millions of
people and t is number of years AFTER 2006. In 2006, Canada’s population exceeded Uganda’s by 4.9 million (t=0
for both equations).
True
Canada
A  33.1 e
 0.009 0
A  33.1e
Canada – Uganda =
0

Uganda
A  33.1
A  28.2e
0.034 0 
A  28.2  e0 
A  28.2
33.1  28.2  4.9
20. Using the equations,
A  33.1e0.009t for Canada and A  28.2e0.034t for Uganda, where A is measured in millions of
people and t is number of years AFTER 2006. By 2009, the models indicate that Canada’s population will exceed
Uganda’s by approximately 2.8 million people.
False
Canada
A  33.1 e
 0.00910
A  33.1e
Uganda – Canada =
0.09

Uganda
A  36.22
A  28.2e
0.03410 
A  28.2  e0.34 
39.62  36.22  3.4
21. Using the models in problems 18 and 19, indicate that in 2013 Uganda’s population will exceed Canada’s.
True
33.1e0.009t  28.2e0.034t
1.17e0.009t  e0.034t
1.17  e0.034t 0.009t
1.17  e0.025t
ln1.17  0.025t
ln1.17
t
0.025
t  6.28
2006  6.28  2012.28
22. Using the models in problems 18 and 19, Uganda’s growth rate is approximately 3.8 times that of Canada.
True
.009*3.8 = .0342
A  39.62