Objectives:
1. To solve exponential
equations
2. To solve logarithmic
equations
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Assignment:
P. 253: 9-20 (Some)
P. 254: 25-46 (Some)
P. 254: 47-66 (Some)
P. 254: 75-88 (Some)
P. 254: 89-102 (Some)
Homework
Supplement
You will be able to
solve exponential
equations
If 5x = 57, then what is the value of x?
Property of Equality of Exponential Equations
If b is a positive real number not equal to 1,
then bx = by iff x = y.
– 5x = 57 iff x = 7
– This means that one way to solve exponential
equations is to make the bases equal so that the
exponents are equal.
1
Solve 3   
9
x
x 3
Solve 9x = 25
This exponential equation cannot be solved
using the handy Property of Equal Exponents.
We need logarithms.
Sometimes you can’t rewrite your equation so that the
bases are the same (or maybe you just don’t want
to). In that case, try one of the following:
1. Take the log
of both sides
–
–
2. Rewrite the equation
in logarithmic form
Both of these are really the same thing
You may have to use the change of base formula
9 x  25
log 9 9 x  log 9 25
Method 1:
Take the log9 of both sides. The base of the
exponential becomes the base of the log.
x  log9 25
Use the Inverse Property to get x on the left.
log 25
x
log 9
Use the Change of Base formula.
x  1.465
9 x  25
ln 9 x  ln 25
x ln9  ln 25
ln 25
x
ln 9
x  1.465
Method 1e:
Take the ln or log of both sides. Don’t worry
about the original base.
Use the Power Property.
Divide. The Change of Base formula happens
automatically
9 x  25
log9 25  x
log 25
x
log 9
x  1.465
Method 2:
Write the original equation in logarithmic
form.
Use the Change of Base formula.
Solve −3e2x + 16 = 5
Solve each equation.
1. 8x – 1 = 323x – 2
2. 2x = 5
Solve each equation.
1. 79x = 15
2. 4e−0.3x – 7 = 13
You have $1000 burning a hole in your pocket.
Your frugal mom has convinced you to put the
money into a savings account that earns 2.25%
APR, compounded continuously. How long will
it take to double your money?
A new car costs $25,000. The value of the car
decreases by 15% each year. Write an
exponential decay model giving the car’s value
y (in dollars) after t years. In how many years
with the value of the car be ½ the original
value?
According to NATO, a movie ticket to see Star
Wars (Episode IV: A New Hope) in 1977 was an
average price of $2.23. In 1997 when Star
Wars was re-released to theaters in
anticipation of the upcoming prequel trilogy,
ticket prices averaged $4.59.
Use the growth model y = Pert to find the annual
growth rate from 1977 to 1997.
You will be able to solve
logarithmic equations
If log2 x = log2 25, then what is the value of x?
Property of Equality of Logarithmic Equations
If b, x, and y are positive real numbers with
b ≠ 1, then logb x = logb y iff x = y.
– log2 x = log2 25 iff x = 25
Solve log4 (2x + 8) = log4 (6x – 12).
Since x = y means bx = by, we can exponentiate
both sides of an equation using the same
base.
– Exponentiate means to raise a quantity to a power
– Notice the two sides of the equation become the
exponents
– Essentially rewriting a logarithmic equation as an
exponential equation
– Check for extraneous solutions (based on domain)
Solve log7 (3x – 2) = 2.
Solve log6 3x + log6 (x – 4) = 2.
Solve each equation.
1. ln (7x – 4) = ln (2x + 11)
2. log2 (x – 6) = 5
Solve each equation.
1. log 5x + log (x – 1) =2
2. log4 (x + 12) + log4 x = 3
Sometimes we want to solve an equation that looks kind of like a
quadratic. Well, just treat it like one.
e2 x  7e x  12  0


Kind of like

ex  3 ex  4  0
ex  3  0
ex  4  0
ex  3
ex  4
x  ln 3
x  ln 4
x  1.099
x  1.386
k 2  7 k  12  0
 k  3 k  4  0
What if the exponential has different bases? Just pick one to
take the log of, the other one will become an exponent.
5
34 x 5  7 2 x
x
4  2 log 3 7
4 x 5
2x
log 3 3
 log 3 7
5
2x
x
4 x  5  log 3 7
4  log 3 7 2
4 x  5  2 x log3 7
5
x
4 x  2 x log3 7  5
4  log 49
log 3
x  4  2log3 7   5
x  10.929
What if the logarithms have different bases, one of which is a
power of the other? Use a change of base—keep the smaller,
get rid of the larger.
log3 2 x  log9 (13x  3)
log 3 2 x 
log 3 2 x 
log 3 (13 x  3)
log 3 9
log 3 (13 x  3)
2
2  log3 2 x  log3 (13x  3)
log 3  2 x   log 3 (13 x  3)
2
2x
2
 (13 x  3)
4 x 2  13x  3
4 x 2  13x  3  0
 4x 1 x  3  0
x  1/ 4
x3
Solve each equation.
Objectives:
1. To solve exponential
equations
2. To solve logarithmic
equations
•
•
•
•
•
•
Assignment:
P. 253: 9-20 (Some)
P. 254: 25-46 (Some)
P. 254: 47-66 (Some)
P. 254: 75-88 (Some)
P. 254: 89-102 (Some)
Homework
Supplement