Chapter 3 Exponential and Logarithmic Functions

3.1 Exponential Functions and Their Graphs

100

Describe the transformations.

f(x)= 2^x

g(x)= -2^(x+5)-4

Reflection in x-axis

Horizontal shift 5 units left

Vertical shift 4 units down

200

Describe domain, range, intercept, and asymptote.

f(x)=e^x

D: (-∞,∞)

R: (0,∞)

y-int: (0,1)

HA: y=0

300

Identify domain, range, asymptotes, and  y-intercept.

f(x)=-3^x+2

D: (-∞,∞)

R: (-∞, 0)

HA: y=0

y-int: (0,-9)

400

$500 is invested at an annual interest rate of 10% compounded biannually. Find the balance after 10 years. 

Approximately $1326.65 after ten years

500

The approximate number of fruit flies in an experimental population after t hours is given by Q(t)=20e^0.03t, where t≥ 0.

a. Find initial number of flies.

b. How large is the population after 72 hours?

a. 20 flies

b. roughly 173 flies

3.2 Logarithmic Functions and Their Graphs

100

Rewrite exponential equation in logarithmic form.

81^1/4=3

log₈₁ 3=1/4

200

Describe domain, range, and asymptote. 

f(x)=ln(x-1)

D: (1,∞)

R: (-∞,∞)

VA: x=1

300

Rewrite the expression using properties of logarithms.

3log₂(1/2)

-3

400

Expand

ln[0.5(x²(y+z)³)/(a⁵b⁷)]

lnx+1.5ln(y+z)-5lna-7lnb

500

Make describe the transformations and determine the domain, range, and asymptote.

 f(x)=3ln(x+6)-4

Horizontal translation 6 units left

Vertical stretch by a factor of 3

Vertical translation 4 units down

D: (-6, ∞)

R: (-∞,∞)

VA: x=-6

3.3 Properties of Logarithms

100

Condense

3ln(x-5)-ln8x

ln[(x-5)³/8x]

200

Evaluate using the change-of-base formula:

log   (4)

   12

log 4/log 12

= 0.56

300

Rewrite the expression in terms of ln 4 and ln 5

ln

    5/64

ln 5

ln 4^3

= ln 5 - 3ln 4

400

Prove (hint: let x=log_au and y=log_av and change to exponential forms)

log_a(uv)=log_au+log_av

a^x=u    a^y=v

uv=a^xa^y

uv=a^x+y

log_a(uv)=x+y

log_a(uv)=log_a(u)+log_a(v)

500

Expand the following logarithmic expression

log_2(4^2* 3^4)

log_2(4^2) + log_2(3^4)

4 + 4log_2(3)

3.4 Solving Exponential and Logarithmic Equations

100

Solve for x:

4^x = 16

log      4^x = log          16

                            4

      4

x = 2

200

Simplify:

e^ln(5x+2)

5x+2

300

Solve for x:

ln(2x-1) = 5

e^ln(2x-1) = e^5

2x-1 = e^5

2x=149.41

x=74.71

400

Solve for x

log        25 = 2

       x

x^2=25

x = 5

500

Solve for x

5(2^3-x)-13 = 100

5(2^3-x)-13 = 100

5(2^3-x) = 113

2^3-x = 22.6

log          2^3-x = log           22.6

                                    2

        2

3-x = 4.5

-x = 1.5

x = -1.5

3.5 Exponential and Logarithmic Models

100

Find the amount of time to double if the investment is compounded continuously. 

Initial Investment: 10,000

Annual Rate = 3.5%

Pe^rt

10,000e^0.035t

20,000 = 10,000e^0.035t

2 = e^0.035t

ln 2 = 0.035t

t = 19.8 years

200

Population = 134.0e^kt (*in thousands*)

t=0 ; 1990

2000 population = 180,000

Find the value of K

180 = 134e^10k

1.34 = e^10k

ln 1.34 = 10k

0.29 = 10k

k=0.029

300

Population in Tilted Towers = 258.0e^kt

t = 0;1990

2000 population = 478,000

a) Find the value of k

b) Find the population in 2010

a) 478 = 258e^10k

    1.85 = e^10k

    ln 1.85 = 10k

    0.62 = 10k

    k = 0.062

b) 258e^0.062(20)

    258e^1.24

    258(3.46)

    891.5

    About 891,500

400

The amount of John Wicks in the real word can be modeled by: y= 663/1 + 72e^-0.547t

Using the model, find the population for the 19th hour 

663/1+72e^-0.547(19)

663/1.0022

661.54 John Wicks

500

The amount of Defaults skins are released to the real world into a city honored for them. The carrying capacity is 1000. This logistic curve can be modeled by p(t) = 1000/1+9e^-0.1656t

After how many months will the population hit 500?

500 = 1000/1+9e^-0.1656t

500+4500e^-0.1656t = 1000

4500e^-0.1656t = 500

e^-0.1656t = 0.11

-0.1656t = -2.21

t = 13 months