# User manual | Objectives: Assignment: To solve exponential P. 253: 9-20 (Some)

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

You will be able to solve exponential equations

If 5

x

= 5

7

, 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 b

x

= b

y

iff x = y.

5

x

= 5

7 iff x = 7

This means that one way to solve exponential equations is to make the bases equal so that the exponents are equal.

Solve

3

x x

3

Solve 9

x

= 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 25

9

x  log 25

9

x  log 25 log 9

x  1.465

Method 1:

Take the log

9 of both sides. The base of the exponential becomes the base of the log.

Use the Inverse Property to get x on the left.

Use the Change of Base formula.

9

x

25 ln 9

x

 ln 25

x  ln 9

 ln 25

x  ln 25 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 log 25

9

x

x  log 25 log 9

x  1.465

Method 2:

Write the original equation in logarithmic form.

Use the Change of Base formula.

Solve −3e

2x

+ 16 = 5

Solve each equation.

1.

8

x

– 1

= 32

3x – 2

2.

2

x

= 5

Solve each equation.

1.

7

9x

= 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 = Pe

rt

to find the annual growth rate from 1977 to 1997.

You will be able to solve logarithmic equations

If log

2

x

= log

2

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 log

b x

= log

b y

iff x = y.

– log

2

x

= log

2

25 iff x = 25

Solve log

4

(2x + 8) = log

4

(6x – 12).

Since x = y means b

x

= b

y

, 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 log

7

(3x – 2) = 2.

Solve log

6

3x + log

6

(x – 4) = 2.

Solve each equation.

1.

ln (7x – 4) = ln (2x + 11) 2.

log

2

(x – 6) = 5

Solve each equation.

1.

log 5x + log (x – 1) =2 2.

log

4

(x + 12) + log

4

x

= 3

Sometimes we want to solve an equation that looks kind of like a quadratic. Well, just treat it like one.

e

2

x

7

e x

12

0

e x

3



e x

4

0

Kind of like

k

2

7

k

12

0

k

3



k

4

0

x

3 0

x

e  3

x  ln 3

x  1.099

x

4 0

x

e  4

x  ln 4

x  1.386

What if the exponential has different bases? Just pick one to take the log of, the other one will become an exponent.

3

4

x

5 log 3

3

4

x

5

7

2

x

log 7

3

2

x

x

5

3

5

4 x   log 7

3

2

x

x

3

2

4

x

 

x

3

5

4

x

x

3

5

x

4

 log 49 log 3

x

3

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.

log 2

3

x

 log (13

9

x

3)

 

2 

(13

x

3) log 2

3

x

 log (13

x

3) log 9

3

4

x

2

13

x

3

4

x

2

13

x

 

0

x

 log (13

3

x

3)

2

4

x

1



x

3

0

3

x

 log (13

3

x

3)

x  1/ 4 x  3 log

3

 

2  log (13

3

x

3)

Solve each equation.

1.

2

2𝑥

− 12 ∙ 2 𝑥

+ 32 = 0

2.

3 𝑥+4

= 6

2𝑥−5

3.

log

2 𝑥 + 1 = log

8

3𝑥

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