Name_____KEY_________________________
Date____________
Honors Algebra 2 Study Guide for Test: Chapter 7
Part 1: Exponential Functions (7.1/7.2):Graph each function. Determine the y-intercept.
Identify if the function represent growth or decay. ** Calculator allowed to get table**
1
5
1.) y  5(0.7) x
2.) y  (4)x
3.) y  3( ) x
3
2
a = 5 b = 0.7 decay
a = 1/3 b = 4 growth
a = 3 b = 5/2 growth
y-intercept: (0, 5)
y-intercept: (0, 1/3)
y-intercept: (0, 3)



Part 2: Logarithmic Expressions and Equations (7.3-7.5):
Write each equation in logarithmic form.
1
4.) 7 3  343
5.) 52 
25



log 7 343 = 3

Write each equation in exponential form.
7.) log 4 64  3
8.) log 8 2 
4 3 = 64
8 1/3 = 2
4 log4 9 = y
log 4 y = log 4 9
Solve each equation.
1
13.) log 4 x 
2
4 ½= x

2 2 (1/2) = x
2=x
6.) 4  8

1
3
 a calculator)
Evaluate each expression.(without
log4 9
10.) 4
11.) log 81 3
log 4 y = log 4 9
y=9

log 5 1/25 = -2
3
2
1
 2
36
6 -2 = 1/36
9.) log 6

12.) log13 169
log 81 3 = y

81 y = 3
3 4y = 3 1
4y = 1
y=1
4
14.) log b 9  2
b2=9

b2=32
b=3
log 4 8 = 3/2
log 13 169 = y

13 y = 169
13 y = 13 2
y=2
15.) log 8 (3y 1)  log 8 (y  5)
3y – 1 = y = 5

3y – y = 5 + 1
2y = 6
y=3
Name_____KEY_________________________
1
1
16.) log 2 y  log 2 27
17.) log 5 7  log 5 4  log 5 x
2
3

log 2 y = log 2 27 1/3
log 2 y = log 2 33 (1/3)
y=3
19.)
log 5 7 + log 5 41/2 = log 5 x
log 5 7 + log 5 2 2 (1/2) = log 5 x
log 5 7 + log 5 2 = log 5 x
log 5 (7)(2) = log 5 x
14 = x
Date____________
18.) log 8 ( x 2  x)  log 8 12
log 8 (x2 + x) = log 8 12
x2 + x = 12
x2 + x – 12 = 0
(x + 4)(x – 3) = 0
x = -4, 3
1
1
log 7 m  log 7 64  log 7 121
3
2
log 7 m + log 7 64 1/3 = log 7 121 1/2
log 3 x  log 3 4  log 3 12
20.)
log 3 x = log 3 12
4
(4) x = 12 (4)
4
x = 48
log 7 m + log 7 4 = log 7 11
log 7 (m)(4) = log 7 11
4m = 11
m = 11
4
Part 3: Common and Natural Logarithms (7.5-7.6)
Solve each equation. Round to 4 decimal places.
21.) 2  53
22.) 6
x log 2 = log 53
3y log 6 = (y – 1) log 8
(x – 5) log 2.1 = log 9.32
3y log 6 = (y – 1) log 8 
3y log 6 = y log 8 – 1 log 8
3y log 6 – y log 8 = - log 8
y (3 log 6 – log 8) = - log 8
y = - log 8______
(3 log 6 – log 8)
y ≈ - 0.6309
(x – 5) log 2.1 = log 9.32
x log 2.1 – 5 log 2.1 = log 9.32
x log 2.1 = log 9.32 + 5 log 2.1
x = log 9.32 + 5 log 2.1
log 2.1
x ≈ 8.0086
25.) 2e  4  1
26.) 4e
2ex=5
e x = 5 = 2.5
2
x
ln e = ln 2.5
ln e x = ln 2.5
x = ln 2.5
x ≈ 0.9163
- 4 e 2x = - 8
e 2x = 2
x

x log 2 = log 53
log 2 log 2
x ≈ 5.7279
4 x7
24.) 3

 4 3x 3
(4x – 7) log 3 = (3x + 3) log 4

4x log 3 – 7 log 3 = 3x log 4 + 3 log 4
4x log 3 – 3x log 4 = 3 log 4 + 7 log 3
x(4 log 3 – 3 log 4) = 3 log 4 + 7 log 3
x = 3 log 4 + 7 log 3
4 log 3 – 3 log 4
x ≈ 50.3008
3y
 8 y1
x5
23.) 2.1
x

2x
 9.32
 15  7
ln e 2x = ln 2
ln e 2x = ln 2
2x = ln 2
x = ln 2
2
x ≈ 0.3466
Name_____KEY_________________________
27.)

ln 3x  5
Date____________
28) ln( x  10)  0.5
e ln (x – 10) = e 0.5
x – 10 = e 0.5
x = e 0.5 + 10
e ln 3x = e 5
3x = e 5
3x = e 5
3
3
x ≈ 49.4711
29.)
ln x  ln 4x 10
ln 4x2 = 10

e ln 4x2 = e 10
4x2 = e 10
x2 = e 10
x ≈ 74.2066
4
x ≈ 11.64872
Express each in terms of common logs. Then approximate its value to four decimal places.
3
30.) log 4 11
31.) log 2 15
32.) log 20 10


log 11 ≈ 1.7297
log 4

Evaluate each expression.
ln12
33) e
34) ln e
eln 12
eln 12
12
ln e 7x
ln e 7x
7x

log 15 ≈ 3.9069
log 2
7x

log 103 = 3 log 10 ≈ 2.3059
log 20
log 20
35.)Write in logarithmic form: e  6
x
ex=6
log e 6 = x
ln 6 = x

A
Part 4: Exponential Growth and Decay (Throughout chapter 7)
36.) A certain strain of bacteria grows from 40 to 326 in 120 minutes. Find k for the growth
formula y=aekt, where t is in minutes.
Exponential growth
a = value at the start = 40
k = rate of growth (when >0) or decay (when < 0)
t = time = 120
y = final value
y = ae k t
326 = 40e k (120)
8.15 = e k (120)
ln 8.15 = ln (e k (120))
ln 8.15 = 120k
ln 8.15 = k
120
326 / 40 = e k (120)
0.0175 = k
37.) Carl plans to invest $500 at 8.25% interest, compounded continuously. How long will it take
for his money to triple?
Interest Compound Continuously
A = Pe r t
A = final amount = 1500
P = Principle, or original amount = 500
r = rate of interest per year = 8.25% = 0.0825
t = time, in years
1500 = 500 e 0.0825 t
3 = e 0.0825 t
1500 = 500 e 0.0825 t
500 500
log e 3 = 0.0825 t
ln 3 = 0.0825 t
0.0825
0.0825
t ≈ 13.3165
t = about 14 years
(needs to be complete year
not partial)
Name_____KEY_________________________
Date____________
38.) There are currently 850 students at the high school, which represents full capacity. The town
plans an addition to house 400 more students. If the school population grows at 7.8% per year, in
how many years will the new addition be full?
Exponential population model
P(t) = P0 (1 + r)t (for increase)
P(t) = P0 (1 – r)t (for decrease)
P0 = initial population
= 850
1250 = 850 (1+0.078)t
1250 = 850 (1.078)t
1250 = 850 (1.078)t
850 850
25 = 1.078t
17
log 25 = log 1.078t log 25 = t log 1.078
17
17
r = rate (decimal)
= 7.8% = 0.078
log (25/17) = t
log 1.078
t = time
t ≈ 5.1348
t = 5 years
39.) Hugo begins a walking program by walking 1 mile per day for one week. Each week thereafter
he increases his mileage by 10%. After how many weeks is he walking more than 5 miles per day?
Exponential growth
A = Final
a = beginning
r = rate
t = time
A = a(1 + r)t
5 = 1(1 + 0.1)t
5 = 1.1t
log 5 = log 1.1t
log 5 = t log 1.1
log 5 = t
log 1.1
t ≈ 16.8863
= 17 weeks
40.) When Emily was 18 months old, she had a 10-word vocabulary. By the time she was 5 years
old (60 months), her vocabulary was 2500 words. If her vocabulary increased at a constant percent
per month, what was that increase?
Exponential growth
A = Final
a = beginning
r = rate
t = time
A = a(1 + r)t
t = 60 – 18 = 42
2500 = 10(1 + r)42
250 = (1 + r)42
42____
42_____
√ 250 = ( √ 1 + r )42
take the 42nd root:
42  MATH  5.  250  enter
1.1405 = 1 + r
-1
- 1___
0.1405 = r
which translates to 14.05% or 14%