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Exponential and Logarithmic Equations
To solve exponential and logarithmic equations
Exponential equation,
Logarithmic equation
cx
Take note: Any equation that contains the form b such as
where the exponent includes a  b cx a variable, is an exponential
equation.
Problem 1: What is the solution of 16 3 x  8 ?
16
3x
2 
4 3x
8
2
Rewrite the terms with a common base
3
2
2
12 x  3
12 x
1
x
4
Your turn
3
Power Property of Exponents
If two numbers with the same
base are equal then the exponents
have to be equal
Solve and simplify
Solving an Exponential Equation- Different Base
Problem 2: What is the solution of 153 x  285 ?
log 15 3 x  log 285
Take the logarithm of each side
3 x log 15  log 285
Power Property of Logarithms
log 285
3x 
log 15
Divide each side by log 15
to isolate the x, then by 3
x  2.087291136 / 3
x  0.6958
Check your answer
Or you can divide the whole expression by 3log15
be careful with ( ) in the graphic calculator
15 30.6958   285
Your turn
5
2x
 130
Answer:  1.5122
Solving an Exponential Equation with a Graph or Table
Problem 3: What is the solution of 4 3 x  6000 ?
Method 1
Solve using a graph
Using a graphing calculator
Y1  4 3 x
Y2  6000
x  2.09
Method 2
Solve using a table
3x
Enter Y1  4 Use table setup
and ΔTbl to locate the x- value
that gives you the y-value closest
to 6000
x  2.09
Adjust the window
that you can find
the point of intersection
Your turn
a )7 4 x  800
b )5.2
3x
 400
Answer:
a )  0.8588
b )  1.2114
Method 1
Method 2
Remember to add the note to set up graphing calculator
Modeling with an Exponential Equation.
Problem 4: Wood is a sustainable, renewable, natural resource when
you manage forest properly. Your lumber company has 1,200,000
trees. You plan to harvest 7% of the tree each year. How many years
will it take to harvest half of the trees?
Know
Number of trees
Rate of decay
Need
Number of years it takes to
harvest 600,000 trees
Plan
Write an exponential equation
Use log to solve the equation.
Step 1: Is an exponential model reasonable for this situation?
Yes, you are harvesting a fixed percentage each year.
Step 2: Define the variables and determine the model
n: the number of year it takes to harvest half of the tree.
T(n): the number of trees remaining after n years.
n
T
(
n
)

a
(
b
)
A reasonable model is
Step 3: Use the model to write an exponential equation
T ( n )  600 ,000
a  1,200 ,000
1,200 ,000( 0.93 )n  600 ,000
r  7%  0.07
b  1  r  1  ( 0.07 )  0.93
Step 4: Solve the equation. Use log
1,200 ,000( 0.93 )n  600 ,000
0.93 n 
600 ,000
1,200 ,000
log 0.93 n  log 0.5
n log 0.93  log 0.5
log 0.5
n
log 0.93
n  9.55
Isolate the term with n
Take logarithm of each side
Power property of Logarithm
Solve for n Use a calculator
It will take about 9.55 years
to harvest half of the original trees
Your turn: After how many years will you have harvested half
of the trees if you harvest 5% instead of 7% yearly.
Answer
T ( n )  600 ,000
a  1,200 ,000
r  5%  0.05
b  1  r  1  ( 0.05 )  0.95
1,200 ,000( 0.95 )n  600 ,000
log 0.95 n  log 0.5
n log 0.95  log 0.5
log 0.5
n
log 0.95
 13.51 years
Take a note: A logarithmic equation is an equation that includes one
more logarithms involving a variable.
Problem 5: Solving a Logarithm Equations.
What is the equation of log ( 4 x-3 )  2?
Method 1 log( 4 x  3 )  2
4 x  3  10 2
4 x  103
Write in exponential form
Simplify and solve for x
x  25.75
Method 2
Your Turn
log ( 3-2 x)  -1
Method 3
Answer: 1.45
Using logarithmic Properties to Solve an Equation
Problem 6: What is the solution of log( x  3 )  log x  1?
log  x  3x   1
 x  3x  101
x 2  3 x  10  0
 x  5  x  2   0
x  5 or x  2
Product Property of Logarithm
Write in exponential form
Simplify to a quadratic equation in
standard form
Factor the trinomial
Solve for x
Check
log( 2  3 )  log( 2 )  1
log( 5  3 )  log( 5 )  1
log 2  log 5  1
0.3010  0.6990  1
Your turn: What is the solution of log 6  log 3 x  2
Answer:
200
6
10 
3x
1
6

100 3 x
3 x  600
x  200
2
CW/HW
Form G