Math IB SL – year 1
Laws of Logarithms – worksheets I and II
Part I:
Exponential Form:
baseexponent = result
Logarithm Form:
log base (result ) = exponent
“The number to which you must raise (base) to get (result)
is (exponent)”
1. Write out each of the following in logarithmic form.
a)
b)
c)
3
4
d)
e) 81 = 27
2. Write out each of the following in exponential form.
a)
b)
c)
d)
3. Evaluate:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
4. Solve for x:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
5. Determine the Value of Each of the Following
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
p)
r)
(
5
)
n) log 2 4 4 − log 3 9
q)
1
3
o)
Part II:
Product Law for Exponents:
Product Law for Logarithms:
Quotient Law for Exponents:
Quotient Law for Logarithms:
Power Law for Exponents:
or
p
Power Law for Logarithms: log a ( N ) = p log a ( N )
or
 1 1
logb  N p  = log b ( N )

 p


1. Express as a single logarithm.
a)
b)
c)
d)
e)
f)
h)
g)
2. Evaluate:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
l)
k)
 9
log 2  4 
4
3 
Laws of Logarithms III
Examples:
1. Solve for x:
2.
Check:
Check:
1+0=1
Solve the Following:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u)
v)
w)
Logarithms IV
1. Solve the equations below. Express your answer accurate to two decimal places.
a)
b)
e)
f)
i)
j)
c)
=
d)
g)
h)
k)
l)
m)
2. The speed of a barge after time, t, in seconds is given by
where
is the speed when the engines are
stopped. Calculate the speed after 3 seconds, if the speed of the barge is 12km.h at the time when the engines are stopped.
3. Radium is used in the making of luminous paint for watch dials. Raphael lost her watch in a construction site. If it were found
8100 a later, what fraction of the original luminosity would be left in the watch dial is the half life of radium is 1620 a?
4. During the transportation of the isotope thorium-243, to a nuclear waste facility, a spill occurred near a populated area. The area
was evacuated until the isotope decayed to its original radioactivity. If the half life of thorium -243 is 24 d, for how long was the
area evacuated?
5. An investment of $100 is made is a term deposit that pays 16%/a compounded annually. The value A, of the investment in dollars
after time, t, in years, in given by
t
...
a) How long will it take for the investment to be worth $150?
b) How long will it take to double the value of the investment?
6. In 1976 a research hospital bought half a gram of radium for cancer research. Assuming the hospital still exists, how much of this
radium will the hospital have in the year 6836 if the half life of radium is 1620 a?
7. The number of cells in a culture grows according to the equation
in seconds. How long will it take the number of cells to grow to
, where A is the number of cells after time, t,
?
8. A cloth, made from plant material was found intact during the excavation of a tomb. The amount of C14 in the cloth was 2.7 g.
The original amount of C14 was 3.8 g. What is the age of the cloth? (½ life =5760 a)
9. Health officials wound traces of radium-F beneath the local library. After 69 d they observed that a certain amount of the
substance had decayed to
of its original mass. Determine the half life of radium-F?
10. Phoebe Small is out driving in her rocket ship. She fills up with fuel and takes off. When she starts the last stage of her rocket,
she is going 4230 miles per hour (mph). Ten seconds later she is going 6850 mph. While the last stage is running, you may assume
that Phoebe’s speed increase exponentially with time.
a) In order to go into orbit, Phoebe must be going 17 500 mph. She took in enough fuel to last for 30
seconds. Will she orbit? Explain.
b) What is the minimum length of time the last stage could run and still get Phoebe into orbit?
c) How long would the last stage have to run to get Phoebe going 25 000 mph so that she could go off
the Moon?
11. Assume that whenever you wash a pair of blue jeans, they lose 4% of the color they had just before they were washed.
a) Explain why the percent of the original color left in the blue jeans varies geometrically with the number
of washings. What is the common ratio?
b) How much of the original color would be left after 10 washings?
c) Suppose that you buya new pair of blue jeans, and decide to wash them enough times so that only
25% of the original color remains. How many times must you wash them?
d) Explain why an arithmetic sequence would not be a reasonable mathematical model for the amount of
color left after many washings?