4.03 Properties of
Logarithms
College Algebra:
One Step at a Time. Page 509-519:
#5,6,13,17,18,23,25,28,29,31,53,54,56,57,58,61,63,64,66.
Dr. Robert J. Rapalje
Seminole Community College
Sanford, FL 32773
To see
Section 4.03, with
detailed explanations, examples, exercises, and answers,
click here!
Preliminary thoughts for problems 1 – 18
Remember the laws of
logarithms:
I. 
II. 
III.
P. 511 # 5.
Solution:
Remember that if you have
a quotient, you SUBTRACT
logarithms. 
Next,
write
as
an exponent 
.
The last
step is to bring down the exponents:
and
This is
the final answer: 
.
P. 511 # 6. 
Solution:
Remember that if you have
a quotient, you SUBTRACT
logarithms. 
Next,
write
as
an exponent 
.
The last
step is to bring down the exponents:
and
This is
the final answer: 
.
P. 513 # 13. 
Solution:
Remember that if you have factors in a product, you must
ADD logarithms, and
if you have a quotient, you SUBTRACT
logarithms. It turns out that for ANY factor that is in the numerator you
must ADD
logarithms, and
for any factor in the denominator you must
SUBTRACT logarithms!
In each
logarithm, bring down the exponents:
P. 513 # 17. 
Solution:
Remember that, as in the previous problems, if you have factors in a
product, you must ADD
logarithms, and if you have a quotient, you
SUBTRACT logarithms. It turns out that for
ANY factors that is in the numerator you must
ADD
logarithms, and for
any factors in the denominator you must
SUBTRACT logarithms!
In the
first logarithm, 
for
any base b is 
,
since 
.
In the
third logarithm, just bring down the exponent.
P. 513
# 18.
Solution:
Remember that, as in the previous problems, if you
have factors in a product, you must ADD logarithms, and if you have a
quotient, you SUBTRACT
logarithms. It turns out that for ANY factors that is in the numerator you
must ADD
logarithms, and for any factors in the
denominator you must SUBTRACT
logarithms!
In the first logarithm, for
any base b is ,
since .
In the second and third
logarithm, just bring down the exponent.
Preliminary thoughts for
problems 19 – 32
Beginning
with problem #19, the laws of logarithms are applied in reverse.
Whereas in
the first few exercises, we wrote the laws this way:
Now, we
are going to use them in this way:
Before
when you had the logarithm of a product, you would write this as the sum of
two logarithms. Now, when you have the sum of two logarithms, you can write
this as the product of a single logarithm. Before, when you had a
quotient, you could write this as the difference of two logarithms. Now,
when you have a difference of two logarithms, you can write this as the
quotient of a single logarithm.
P. 514
# 23.
Solution:
Beginning with problem
#19, the laws of logarithms were applied in reverse. In #1-19, the
exponents were brought down to become coefficients. Now the coefficients
will be moved up to become exponents.
Now, logarithms with
positive coefficients become factors in the numerator of the fraction, and
negative coefficients will become denominator factors. It may help to
re-write the expression with the positive terms first, although this step is
not really necessary if you understand it.
P. 515 # 25.
Solution:
Here the
laws of logarithms are applied in reverse. In #1=19, the exponents were
brought down to become coefficients. Now the coefficients will be moved up
to become exponents.
Now, logarithms with positive
coefficients become factors in the numerator of the fraction, and negative
coefficients will become denominator factors.
or 
P. 515 # 28. 
Solution:
First, by distributive property, you can write this:
Now, the
coefficients become exponents:
Now,
express as a single logarithm. Remember that terms with negative
coefficients become factors in the denominator.
Beginning
with problem #19, the laws of logarithms were applied in reverse. In #1-19,
the exponents were brought down to become coefficients. Now the
coefficients will be moved up to become exponents.
Now,
logarithms with positive coefficients become factors in the numerator of the
fraction, and negative coefficients will become denominator factors.
or
P. 515 # 29.
Solution:
First, by
distributive property, you can write this:
Now, the
coefficients become exponents:
It may be
helpful to re-write this with the positive terms first:
Now,
express as a single logarithm. Remember that terms with negative
coefficients become factors in the denominator.
P. 516 # 31. 
Solution:
First, by distributive property, you can write this:
Now, the
coefficients become exponents:
Now,
express as a single logarithm. Remember that terms with negative
coefficients become factors in the denominator.
P. 517 # 53.
Solution:
This is a “quickie”! This really means
,
and since log base e
is the inverse of
e raised to the power,
these are inverse functions
of one another! Therefore, the answer is
the power,
which is 
.
Note: Since this is an “ln” problem, you
can also use the calculator to calculate the answer using the
LN button.
P. 517 # 54.
Solution:
This is an even quicker “quickie”! Doesn’t
equal
?
Then #54
is exactly the same problem as
#53.
You can also use the calculator with the
LN
button, and the answer should also be
.
P. 518 #
56. 
Solution:
Remember that if you have factors in a product, you must
ADD logarithms,
and if you have a quotient, you
SUBTRACT logarithms. It turns out that for
ANY factor that is in the numerator you must
ADD
logarithms, and for any factor in the
denominator you must SUBTRACT
logarithms!
In the
second logarithm, you have the same base for the logarithm as the base
number of the exponent, 
,
so the answer is the power,
which is 
.
In the
first and third logarithm, you have different base numbers, so the only
thing you can do is bring down the exponent.
and
This is
the final answer: 
.
P. 518 #
57. 
Solution:
Remember as before that if you have factors in a product, you must
ADD
logarithms, and if you have a quotient, you
SUBTRACT
logarithms. It turns out that for ANY factor that is in the numerator you
must ADD
logarithms, and for
any factor in the denominator you must
SUBTRACT logarithms!
In the
first logarithm, there is nothing you can do, at least for now. At the end
of the next lesson, you will be able to find the value, but for now, just
leave it as it is.
In the
second logarithm, you have different base numbers, so the only thing you can
do is bring down the exponent. 
.
In the
third logarithm, you have the same base for the logarithm as the base number
of the exponent, 
,
so the answer is the power,
which is 
.
This
is the final answer: 
.
P. 518 # 58.
Solution:
Remember as before that if you have factors in a product, you must
ADD
logarithms, and if you have a quotient, you
SUBTRACT logarithms.
It turns out that for ANY factor that is in the numerator you must
ADD
logarithms, and for any factor in the
denominator you must SUBTRACT
logarithms!
In the
first logarithm, you have the same base for the logarithm as the base number
of the exponent, 
,
so the answer is the power,
which is 
.
In the
second logarithm, you have different base numbers, so the only thing you
can do is bring down the exponent.
.
In the
third logarithm, there is nothing you can do, at least for now. At the end
of the next lesson, you will be able to find the value, but for now, just
leave it as it is.
This is
the final answer: 
.
P. 519 # 61. 
Solution:
As
in # 58,
remember that for factors in the numerator you must
ADD logarithms,
and for factors in the denominator you must
SUBTRACT logarithms!
Notice
that in the first two logarithms, you have the same base number as the base
of the exponent, so as before, the answer is the
power.
The answers are 
and
respectively.
In the
third logarithm, the base number is NOT the same, so you can’t simplify
it—at least not until the next section!!
The final
answer is: 
or
or
.
P. 519 # 63.
Solution:
Before
beginning this problem, remember that logarithms and exponential functions
with the same base numbers are inverses one of another. This is to say that
, where the base number is b>0 and not equal to 1. As
examples, 
, 
, 
, or 
. The logarithm, base b, of b raised to a
power
is the power!
Now,
since this is log base 7,
it would be nice if you could express the rest of the problem with a base
number of 7.
Remember that 
is actually 
, 
is actually
, and when you multiply with the same base number, by the
law of logarithms, you add
exponents! So,
(Remember
that 
)
Since the base numbers of the logarithm and
the exponential are both the same (that is, 7), the final answer is the exponent
P. 519 # 64.
Solution:
Continue with the same way of
looking at logarithms in this next example. Notice that the base of the
logarithm is 
, and that the numbers
and 
can both be expressed as powers of
.
(Remember that
?)
, since the base numbers of the logarithm and the
exponential are both 5,
the final answer is the exponent
.
P. 519 # 66. 
Solution:
As
in previous problems,
remember that for factors in the numerator you must
ADD
logarithms, and for
factors in the denominator you must SUBTRACT
logarithms!
Notice
that in the first logarithms, there is nothing to do, since you do NOT have
the same base number as the base of the exponent. However, in the last two
terms, you DO have the same base number as the base of the exponent, so the
answer is the power. These exponents are
and
respectively.
The final
answer is: 
or
.
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