Math 331- Algebra II
Chapter 12 Notes
Name_______________________________
12.1- Exponential Functions
Def- An exponential function with base b is defined by f ( x)  b x or y  b x , where b is a positive
constant other than 1 where b>0, b  1, and x is any real number.
Ex- f ( x)  7
x
1
g ( x )   
8
x
h( x )  2
6 x 3
2
r ( x )  5 
7
x
Ex- Evaluate the exponential function for x = -3 and x = 5. Round your answer to the nearest
hundredth.
1
b) g ( x)  32.8 
2
a) f ( x)  12(7) x
x 3
Graphs of Exponential Functions
Exponential Decay
Exponential Growth
x
y  b for 0<b<1
y  b x for b>1
Natural Exponential Function
y  ex
Asymptote:
Points:
Domain:
Range:
Asymptote:
Points:
Domain:
Range:
Asymptote:
Points:
Domain:
Range:
Ex- Graph each function by making a table of coordinates.
a) f ( x)  4  x
b) f ( x)  4 x
1
c) h( x)   
3
x 1
d) g ( x)  e x  1
Formulas for Compound Interest
After t years, the balance, A, in an account with principal P and annual interest rate r is given by the
following formulas:

1. For interest compounded n times per year, A  P1 

rt
2. For interest compounded continuously, A  Pe .
nt
r
 .
n
Ex- What percent interest is needed for an investment of $1000 to increase to $1500 in one year
compounded monthly?
Ex- A house was purchased in 1996 for $150,000 and sold 20 years later at an undisclosed amount.
If the percent appreciation is 4.2% over the 20 year period, find the cost of the home.
12.2- Logarithmic Functions
Note: The inverse function of the exponential function with base b is a logarithmic function,
y  log b x . The inverse function of the natural exponential function with base e is a natural
logarithmic function, y  ln x .
Def- For x>0, b>0, and b  1, y  log b x is equivalent to b y  x . The function y  log b x is the
logarithmic function with base b.
Def- For x>0, y  ln x is equivalent to e y  x . The function y  ln x is called the natural logarithmic
function.
Exponential Form
by  x
(for b>0 and b  1)
Logarithmic Form
y  log b x
ey  x
y  ln x
Ex- Write each equation in its equivalent exponential form.
a) log 7 49  2
b) log m p  q
e) ln e  1
d) ln 5 = 1.6094
c) log 5 125  3
f) ln 4 = p
ex- Write each equation in its equivalent logarithmic form.
1
a) 3 4  81
b) e m 1  n
d) 4 m  n
e) e 8  k
c) 8 3  2
Properties of Logarithms
log b b  1
log b 1  0
ln e  1
ln 1  0
log b b x  x
ln e x  x
b logb x  x
e ln x  x
Ex- Evaluate without using a calculator.
a) log 7 1
b) ln e 4
c) log 10000
f) 4 log4 9
g) ln e
h)
e ln x
d) e ln 7
3
i) log 2 8 x
e) log 4 64
Graphs of Logarithmic Functions
Logarithmic Function with Base b
for x>0 and b>0
y  log b x
Natural Logarithmic Function
for x>0
y  ln x
Asymptote:
Points:
Domain:
Range:
Asymptote:
Points:
Domain:
Range:
Ex- Graph each logarithmic function.
a) g ( x)  log 3 x
b) g ( x)  ln x  1
To find the domain of a logarithmic function without graphing:
For y  log b x and y  ln x , set x>0 and solve. Write your answer in interval notation.
Ex- Find the domain without graphing.
a) r ( x)  7 ln( x  3)  5
b) m( x)  log 2 (4 x  7)  9
c) k ( x)  ln( 5  3 x)
d) p( x)  log( x  5) 2
Ex- The formula f ( x)  38 log( x  2)  13 represents the diameter (in inches) of a tree x years after a
fire.
a) Find the diameter of a tree 15 years after a fire.
b) If a tree has a diameter of 70 inches, how many years has it been since the fire occurred?
12.3- Properties of Logarithms
Properties of Logarithms
For any values of M and N such that M>0 and N>0, the following rules apply:
log b (MN )  log b M  log b N
ln( MN )  ln M  ln N
Product Rule:
Quotient Rule:
M 
log b    log b M  log b N
N
M 
ln    ln M  ln N
N
Power Rule:
p log b M  log b M p
p ln M  ln M
Ex- Express as a single logarithm, then simplify.
a) log 7 5  log 7 8
b) 3 log m 4  7 log m 2
d) ln( m  4)  ln( m 2  16)
e) log 3 27  log 3 15  log 3 18
p
c) ln x  4 ln y  6 ln z
f) log( 2000)  log( 50)
g) ln 6  ln 15  ln 20
Ex- Use properties of logarithms to expand each expression as much as possible.
 3 x2 
 3x 

a) log 2  
b) ln m  5
c) log 5 


w
x

4
 


 x2 y4 
d) ln  10 3 
z k 
Ex- State whether each is true or false, then explain.
log 5 7
a)
b) log 3 9  log 3 81  2
 log 5 7  log 5 9
log 5 9
c) ln 11  ln 10  (ln 11)(ln 10)
When trying to calculate something like log 5 8 on the calculator, you need to use the change of base
property.
Change of Base Property
For any logarithmic bases a and b and any positive number M, log b M 
Ex- Use the change of base property to evaluate each of the following.
a) log 4 17
b) log 2 24
c) log 7 8
log a M
ln M
or log b M 
.
ln b
log a b
d) log .9 16.2
12.4- Exponential and Logarithmic Equations
Solve each equation. Give exact solutions only. (Exact solutions means…do not use a calculator.)
**** You can only take the log or ln of positive number. Check every single answer if the original
problem contains log or ln.
1. log 9 x  log 9 4
2. ln( x  2)  4
3. log( x  4)  3
4. ln( 5 x)  4
2x
5. 23 x  4 x
6. 8 x  16 2
1
7.  
3
8. ln 5  ln t  4
9. 2 4 x  174
10. log 8  log( 7 x)  log 9
11. log 6 x  log 6 ( x  1)  1
12. 5 ln( 7 x)  3
13. 7e 2t  21
 81
14. log(  x)  log( 3x  2)  log 8  0
15. 18  1.7 x  0
16. log( 5 x)  log( 8 x  6)
17. ln x  ln( x 2  4 x)  ln 2 x
18. ln( x  3)  7  ln( 2 x)
19. 7 2 x  3 x 1
Ex- Kim invested $500 in an account that is compounded continuously. How long will it take for the
investment to reach $600 at 7% interest?
Ex- Carmen invests $20,000 in an account that gains 2% interest compounded semi-annually. How
long will it take her to double her investment?