Mathematical Investigations III
Name:
Mathematical Investigations III
Logarithms
Review
1.
 125  625 
The value of log 5 
 is
 25 
2.
If log10  m  b  log10  n  , then m =
If y  log a  x  and a > 1, then which of the following statements is (are) correct?
a.
If x = 1, then y = 0.
b.
If x = a, then y = 1.
c.
If x = 1 , then y is imaginary.
d.
If 0 < x  1, then y  0 and decreases without limit as x approaches 0.
3.
4.
Solve for z: log10  z  15  log10  z   2
5.
a.
101og10 (7) 
b.
log 3  317  
c.
103 log(4) =
d.
log 3

6.
 3=
5
4
If you are given log  8 = t and log  9  = r, give the values of each of the following
without using either tables or a calculator. [If this cannot be done, write "Impossible"]
4 
b. log  =
5 
a. log 17  =
c. log 15 =

d. log  600  =
Logs 14.1
Rev. S12
Mathematical Investigations III
Name:
7.
Eliminate the logs to solve for p in terms of q: log(p) + log(q) = log(p + q).
8.
Simplify
9.
Consider the graphs of y = 2 log(x) and y = log(2x). Without using your graphing
calculator, which of the following is(are) true?
a. They do not intersect.
b. They intersect in only 1 point.
c. They intersect in only two points. d. They coincide.
e. They intersect in a finite number of points, but more than twice.
10.
log b  8 
1
log b  
8
=
a  ar n
Solve for n in terms of s, a, and r: s 
1 r

11.
Solve each for x:
a.
12.
  3
ln x3
3
b.
ln  x 2   2ln
 x  1
If a 0.3  x , then log x  a  =
Logs 14.2
Rev. S12
Mathematical Investigations III
Name:
13.
Write the following expression as a single log:
2 log  w  
1
2
log  x   log  y   3log  z  
4
3
14.
Solve for x:
log2  6 x  5  log2  x   2
15.
Simplify:
16.
Solve for x:
9 x  2  240  9 x
17.
Solve for x:
log  x 2  3 x  6   1
log  x  4  log  x  1  log  6 
5m 4n
b
=
bm 2n 

bm 2n 

Logs 14.3
Rev. S12
Mathematical Investigations III
Name:
18.
Write as a log of a single expression:
a.
log  b 2  9   3log  a   2 log  b  3  log
b.
log  a   2 log  b  
3
log  c   log
2
19.
To do a bone scan in the hospital, the patient is injected with a radioactive substance with
a half-life of approximately 6 hours. There is a calcium additive to make the substance
adhere to the bones, which then become visible when pictures are taken. After how many
hours will the radioactive substance decay to 10% of the original amount?
20.
Kinshasa, the capital of the Congo, had a metropolitan area population of approximately 4
million people in 1993. It is estimated that the population is growing at a rate of almost
10% per year. Assuming this rate, how long did it take the population to double? How
many years did it take for the population to reach 20 million?
Logs 14.4
Rev. S12
Mathematical Investigations III
Name:
21.
22.
The half-life of radioactive radium is approximately 1600 years.
a.
If a sample initially contains 20 grams of radium, how much radioactive will be
left after 700 years?
b.
After how many years will 60% of a sample have decayed?
Solve for x: (Give exact answers whenever possible.)
a.
log  x   log 14  log 3  log  2 
b.
log  x   log  48  log  3
c.
log  x   2  10  x
d.
log2  2log3  x  4   3
Logs 14.5
Rev. S12
Mathematical Investigations III
Name:
23.
24.
Given the graphs of any of the following, you should be able to write the equation (See
problem 25). Here, you are given the equation. Graph the functions without using your
calculator.
a.
y  log 2  x 
b.
y  log2   x 
c.
y   log2  x 
d.
y  4log 2  x 
e.
y  log2  x  2
f.
y  log 2  x 
g.
y  log2  x   2
h.
y  log  x
Prove each of the following:
a.
Given log(ab) = log(a) + log(b), b.
a
derive log    log  a   log  b 
b
25.

Given: a b  c ,
log d  c 
derive: b 
log d  a 
For each of the following, find a function of the form y   log a (x  b)  c :
Logs 14.6
Rev. S12