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