1.
MATH 1314 - REVIEW - TEST 4
Match each graph with the correct equation.
i) y  logb x with b  1
ii) y  logb x with 0  b  1
iii) y  b x with b  1
(a) ______
iv) y  b x with 0  b  1
(b) _________
y
y
x
x
(c) ______
(d) ______
y
y
x
x
2.
Graph the following functions by plotting at least 4 accurate points. Be sure to draw and label any
asymptotes. Also, for each graph find:
i) asymptote
ii) domain
iii) range
x
1
b) g  x       4
c) h  x   log 2 x
3
Evaluate. Round to 4 decimal places:
a) e 
b) e 2 
c) e0.017(15) 
d) 3 e 
128
 0.07 
0.07 8
e) e 2 3 
f) 5000 1 
g) 5000e   


12 

ln  2 

h) log 7  4 
i) ln 5 
j) log  5 
k)
0.05
5
ln  
6 
l)
0.035
Change to logarithmic form:
x3
a) f  x   2   2
3.
4.
a) 3 2 x  10
5.
c) e x  y  9
3
d)  
4
x 5
 91
Change to exponential form:
3
c) log b    8
7
5
Find the exact value of the following without using a calculator:
 1 
a) log 3  
b) logb 4 b
c) 2 log2 59
 27 
a)
6.
b) 4 3  64
ln  x  6  7
b) log 1 x  2 y  6
d) log s  r  t
d) log 4 32
e) log 6 0
g) eln 23
f) log 1 1
h) log a a
i) log  9
9
7.
Find the domain of the following functions and write the answer in interval notation:
a) f  x   log6  x  3
b) h  x   log  2 x  9 
c) j  x   log 1 1  x 
3
8.
Write each expression as a sum and/or difference of logarithms. Do not leave any radicals or powers
in the answer.
 x3 x  1 
 25 x 4 
3
 3x


log
a) log 5  5 3 
b) ln x 4 y  5
c) log b 
d)


2
5
2
w z 
  x  1 
1000 y z 


9. Write each expression as a single logarithm. Do not leave any negative or fractional exponents in the
answer.
1
1
a)  log x  log y 
b) 2 log 2 x  log 2 y
2
3
1
c) 2 ln  x  3  ln  x  2   4 ln x  ln y
2
10. Use the change of base formula and a calculator to evaluate (to 4 decimal places):
a) log 12 4000
b) log 1 7


2
11. Solve the following equations. Find the exact answer, and if necessary use your calculator to give a
decimal approximation to 3 decimal places.
a. 5
x 3
 25
x
x 5
b.
c. log 4 x  3
e. ln 5 x  ln( 2 x  1)  ln 4
d.
f.
 1
x 10
  8
32
 
ln e x 3  5
2 log 2 ( x  1)  log 2 4
e x 1  4
g. log 1 4 x  log 1  x  3  log 1 2
h.
i. log3  x  3  1 log3  x  5
j. 5 x 3  210
k. 6 x  3  4 x
m. log4 x  log4  x  6  2
l. 2 x 1  51 2 x
n. 4 x  137
p. 9e x  5
2
2
2
12. How long will it take an investment to double if it is invested in a certificate of deposit that pays 7.84%
annual interest rate compounded continuously. Round to the nearest tenth of a year.
13. The number of bacteria, N(t), present in a culture at time t hours is given by
N  t   10, 000 e0.1 t
a) What is the initial number of bacteria?
b) How many bacteria are present after 4 hours?
c) When will there be 50,000 bacteria present?
d)
14. Radioactive strontium decays according to the function
A(t )  5e0.0239 t
where t is time in years, and A(t) is the amount in grams.
a) What is the initial amount of radioactive strontium present?
b) How many grams will be present after 20 years?
Answers 1. a) i
2.
b) iv
c) iii
d) ii
x
1
b) g  x       4
3
i) horizontal asymptote y  4
ii)   , 
iii)   , 4
x3
a) f  x   2   2
i) horizontal asymptote y  2
ii)   , 
iii)   2, 
y

y=4





   






x
   







y = -2





y
x





c) h  x   log 2 x
i) vertical asymptote x = 0
ii)  0,   iii)   , 




y
x=0
x
   






3.
4.

a) 2.7183
e) 82.6168
i) 0.8047
k) 13.8629
a) 2 x  log 3 10
5.
a) e7  x  6
6.
a)  3
f) 0




b) 0.1353
c) 1.2905
f) 8739.1323
g) 8753.3625
j) undefined as 5 is not in the domain of
l) 5.2092
b) 3  log 4 64
c) x  ln  y  9 
1
b)  
7
1
4
g) 23
b)
2 y 6
x
c) 59
h) 1
d) 1.3956
h)  3.3804
log x
d) x  5  log 3 91
3
c) b 8   
d) 10r t  s
5
5
d)
e) undefined
2
i) 9
4
  3, 
9

b)  ,  
2

7.
a)
8.
a) 2  4 log 5 x  5 log 5 w  3 log 5 z
1
c) 3logb x  logb  x  1  2 logb  x  1
2
9.
a) log xy
3 y 
b) log 2  2 
 x 
c)
  , 1
1
3


b) 3  ln x  ln  y  5   or 3ln x  ln  y  5 
4
4


1
d) log x  3  2 log y  5log z
3
  x  32 x 4 

c) ln 
 y x  2 
b)  2.8074
15
4
11. a) 13
b) 
c) 64
d) 8
e)
f) 3
3
4
g) 3
h) ln 4  1 or 2.386
i) 2
ln 210  3ln 5
ln 210
3 ln 6
3log 6
or
 3 or 0.322
j)
k)
or
or 13.257
ln 5
ln 5
ln 4  ln 6
log 4  log 6
ln 5  ln 2
l)
or 0.234
m) 8
ln 2  2 ln 5
ln 137
5
n)
or 3.549
p) ln   or  0.588
ln 4
9
12. 8.8 years
13. a) 10,000
b) 14,918
c) 16.09 hours
14a) 5 grams
b) 3.10 grams
10. a) 3.3378