PreCalculus 4/2/13

Obj: SWBAT solve exponential and logarithmic equations

and exponential growth and decay functions.

Bell Ringer: Get folder and put name on it. Get scantron

and break packet. Put scantron and packet in your

folder. Explanation to increase grade

HW Requests:

Homework:

pg 296 #11,15, 17, 21, 29, 31, 33, 39, 40, 44 pg 342 #1,3

Read Section 4.1

Announcements:

Take Chapter 3 Test if missing

Tutoring today 3-3:15

4/2/13 Explanation to increase grade

Can increase break project original grade by as much as

25%. Over the next few weeks we will be having ACT Bell

Ringers. You are to complete in pencil the bell ringer in 2

minutes. Then we will take 2-4 minutes to make

corrections in ink. You will get classwork credit for these

assignments. Tardy no credit for that day.

However, your project grade on ACT prep material will be

increased by the percentage of complete bell ringers you

have in your folder. If you complete all assignments then

your grade can increase by 25%

Homework:

pg 296 #11,15, 17, 21, 29, 31, 33, 39, 40, 44 pg 342 #1,3

Read Section 4.1

Announcements:

Take Chapter 3 Test if missing

Tutoring today 3-3:15

Growth Decay

If a quantity increases by the same proportion r in each unit

of time, then the quantity displays exponential growth and

can be modeled by the equation

y C rt ( )1

where

C = initial amount

r = growth rate (percent written as a decimal)

t = time

(1+r) = growth factor where 1 + r > 1

You deposit $1500 in an account that pays 2.3% interest

compounded yearly,

1) What was the initial principal (P) invested?

2) What is the growth rate (r)? The growth factor?

3) Using the equation A = P(1+r)t, how much money would

you have after 2 years if you didn’t deposit any more money?

3) A P r

A

A

t

( )

( . )

$ .

1

1500 1 0 023

1569 79

2

1) The initial principal (P) is $1500.

2) The growth rate (r) is 0.023. The growth factor is 1.023.

If a quantity decreases by the same proportion r in each unit

of time, then the quantity displays exponential decay and

can be modeled by the equation

y C rt ( )1

where

C = initial amount

r = growth rate (percent written as a decimal)

t = time

(1 - r) = decay factor where 1 - r < 1

y C rt ( )1

You buy a new car for $22,500. The car depreciates

at the rate of 7% per year,

1) What was the initial amount invested?

2) What is the decay rate? The decay factor?

3) What will the car be worth after the first year?

The second year?

1) The initial investment was $22,500.

2) The decay rate is 0.07. The decay factor is 0.93.

3 1

22 500 1 0 07

20 925

1

) ( )

, ( . )

$ ,

y C r

y

y

t

y C r

y

y

t

( )

, ( . )

$ .

1

22 500 1 0 07

19460 25

2

1) Your business had a profit of $25,000 in 1998. If the profit increased

by 12% each year, what would your expected profit be in the year

2010? Identify C, t, r, and the growth factor. Write down the

equation you would use and solve.

2) Iodine-131 is a radioactive isotope used in medicine. Its half-life or

decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131,

how much would be left after 32 days or 4 half-lives. Identify C, t, r,

and the decay factor. Write down the equation you would use and

solve.

3) Ray Industries bought a touch screen monitor for $1500. It is

expected to depreciate at a rate of 17% per year. What will the value

of the monitor be in 5 years? Round to the nearest dollar.

4) The Smiths bought an apartment for $100,000. Assuming that the

value of the apartment will appreciate at most 5% a year, how much

will the apartment be worth in 7 years?

C = $25,000

t = 12

r = 0.12

Growth factor = 1.12

y C r

y

y

y

t

( )

$ , ( . )

$ , ( . )

$ , .

1

25 000 1 0 12

25 000 112

97 399 40

12

12

1)Your business had a profit of

$25,000 in 1998. If the profit

increased by 12% each year, what

would your expected profit be in

the year 2010? Identify C, t, r, and

the growth factor. Write down the

equation you would use and solve.

C = 25 mg

t = 4

r = 0.5

Decay factor = 0.5

y C r

y mg

y mg

y mg

t

( )

( . )

( . )

.

1

25 1 0 5

25 0 5

1 56

4

4

1) Iodine-131 is a radioactive isotope used in

medicine. Its half-life or decay rate of 50% is

8 days. If a patient is given 25mg of iodine-

131, how much would be left after 32 days or

4 half-lives. Identify C, t, r, and the decay

factor. Write down the equation you would

use and solve.

C = $1500

t = 4

r = 0.17

Decay factor = 0.83

3) Ray Industries bought a touch

screen monitor for $1500. It is

expected to depreciate at a rate of

17% per year. What will the value

of the monitor be in 5 years?

Round to the nearest dollar.

𝑦 = 𝐶(1 − 𝑟)𝑡 𝑦 = $1500(1 − 0.17)5

𝑦 = $1500(0.83)5

𝑦 = $590

C = $100,000

t = 7

r = 0.05

Growth factor = 1.05

4) The Smiths bought an apartment

for $100,000. Assuming that the

value of the apartment will

appreciate at most 5% a year, how

much will the apartment be worth

in 7 years?

𝑦 = 𝐶(1 + 𝑟)𝑡 𝑦 = $100,000(1 + 0.05)7

𝑦 = $100,000(1.05)7

𝑦 = $140,710.04

7 + 32x-1

= 154 -7 -7

32x-1

= 147

log32x-1

= log147 (2x – 1)log3 = log147

2xlog3 – log3 = log147 +log3 +log3

2xlog3 = log147 + log3 2log3 2log3

x ≈ 2.7712 Solving Exponential and

Logarithmic Equations

ex = 12

lnex = ln12

xlne = ln12

x = 2.4849

2ex = 3

ex = 1.5

x = 0.4055

2 2

lnex = ln1.5

Solving Exponential and

Logarithmic Equations

lnx = 14

e14

= x 1202604.284 ≈ x

2lnx = 9

ln x = 4.5

2 2

e4.5

= x

90.0171 = x

Solving Exponential and

Logarithmic Equations

5x-2

= 33x+2

log5x-2

= log33x+2

(x – 2)log5 = (3x + 2)log3

x(log5) – 2log5 = 3x(log3) + 2 log3

xlog5 – 3xlog3 = 2log3 + 2log5

x(log5 – 3log3) = 2log3 + 2log5

(log5 – 3log3) (log5 – 3log3)

x ≈ -3.2116

Solving Exponential and

Logarithmic Equations

32x-1

= 0.4x+2

log32x-1

= log0.4x+2

(2x – 1)log3 = (x + 2)log0.4

2xlog3 – log3 = xlog0.4 + 2log0.4

2xlog3 – xlog0.4 = 2log0.4 + log3

x(2log3 – log0.4) = 2log0.4 + log 3

(2log3 – log0.4) (2log3 – log0.4)

x ≈ -0.2357

Solving Exponential and

Logarithmic Equations