4.01
Definition of
Logarithms
College Algebra:
One Step at a Time. Page 493-501:
#18, 60, 67-69.
Dr. Robert J. Rapalje
Seminole State College of Florida
Altamonte Springs Campus
To see
Section 4.01, with
detailed explanations, examples, exercises, and answers,
click here!
P. 497 # 18.
Solution:
There are
two ways to solve this problem depending upon how familiar you have become
with logarithms. In the first method, set the logarithm equal to x, and
translate this from logarithmic notation to exponential notation:
Next,
translate from radical form to exponential form.
Now, since
the base numbers are the same, the exponents must be equal, so
As a
second method, as you become a bit more familiar with logarithms, there is a
short-cut. Just convert from radical to exponential notation:
Now, since
the base of the logarithm is the same as the base of the exponent, and since
a logarithm is really “the exponent”, then the answer is “the exponent.”
The “log base 3” is the inverse of the operation of “raising 3 to the
power,” so the answer is the “power”, which is
P. 500 # 60.
Solution: First,
translate this from logarithmic notation to exponential notation:
Notice
that this equation has a base number of b which is raised to a power. It
would be nice to end up with b raised to the 1 power. In order to do this,
you could raise both sides to the
power,
which when you multiply exponents,
will give you what you need! It looks like this:
Now, do
you remember how to simplify a negative fractional exponent? Remember that
the denominator gives you the index of the radical, and the numerator gives
you the exponent. In this case, take the square root of 9, and raise to the
-3 power.
You can
also calculate fractional exponents with the calculator, but remember to
place parentheses around the exponent!
P. 501 # 67.
# 68. 
# 69. 
Solution: Since
this is a log base 10
problem, you can solve it with
a calculator. Just use the LOG
button on the calculator. The
answer to #67 is -1, #68 is -2, and #69 is -3.
Just for
fun (math is fun,
isn’t it??), do the
following problems with log base 10 of the calculator, and see what you get
for answers:
The answers should be as
follows: 1, 2, 3, 4, 6, -2, -3, -6,
0, and, of course,
is
Undefined!!
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