4.04 Solving Exponential
Equations
with Logarithms
College Algebra:
One Step at a Time.
Page 523:
#3, 16, 17, 19, 23, 27, Extra Prob, 29, 33, 34, 35, 36, Extra Prob.
Dr. Robert J. Rapalje
Seminole Community College
Sanford, FL 32773
To see
Section 4.04, with
detailed explanations, examples, exercises, and answers,
click here!
Solving equations
using logarithms is not nearly as hard as you think it is, especially if it
is explained in living color! Consider these exercises from pages
520-523.
Notice how the colors make the exercises so much easier to follow. Can
you imagine what this would look like in black and white? Most of our
television is in color--why not math? Have your graphing calculator
ready, especially if you have a TI 84 or a TI 83+!
P. 521 # 3. Three Solutions for
First solution:
Laws of exponents--a "contrived" problem.
Begin by noticing that this is
a very “special” problem in which both sides of the equation can be written
as a power of the same number 3. In mathematics we call this a “contrived
problem,” one that we made up especially because of this special property,
and therefore the method is of little or no use to real life problems, where
things usually don’t come out even.
Notice that
and 
, and substitute these into the original problem:
Remember that when you raise a
power to a power, you multiply exponents:
Notice that the base number on
both sides of the equation is the same. I call it the
“This
Equals
That
Theorem”:
If
, then 
Solve the equation by
subtracting
4x
and
6 from each side:
Divide both sides by -1:
Second solution: Using Logarithms.
How would you solve an
exponential equation if it does NOT have the same base number on both sides
of the equation?
Take the ln of both sides of
the equation:
Using the second law of
logarithms:
Using the distributive
property:
Get all x terms on the left
side by subtracting 
and
from both sides of the equation :
Get the x
variable in one place by factoring out the common factor of x:
Solve for x by dividing each
side of the equation by 
In the final step, parentheses
are included that would be suitable for calculating the answer with a TI 83
or TI 84. To calculate with a TI 85 or TI 86, use parentheses as indicated
in the previous step. Notice that for ALL calculators, you must have
parentheses around the numerator and the denominator. Notice also that at
the beginning of the numerator, it is a NEGATIVE of the ln 9, but the other
two signs in the problem are MINUS (not negative!) signs. Also, for the TI
83/84 calculator, be reminded that there must be a DOUBLE closed parentheses
at the end of the numerator!
Calculate with your calculator:
x = 8
Third solution: Using a Graphing
Calculator—TI84 (or TI83+)
First set the equation equal to
zero:
Since the exponents are
binomials, be sure to put parentheses around each of them, and write this
equation in the form y1= for graphing purposes. Graph this equation with
the graphing calculator. The standard window will be appropriate.
You can probably see that the
answer this problem is the root (zero) that occurs at x = 8. However, you
can’t be sure from looking at it that the answer is exactly 8. Also, most
of the time, the answer will NOT come out even for these types of problems,
so it may be helpful to go ahead and find the zero. With a TI 83/84, use
the keystrokes [2nd] [CALC F4] and look for [2: zero] as
illustrated below.
You will be asked for a “left
bound”, so give the calculator a value to the left of what you think the
zero is, let’s say any number between zero and 7 inclusive, and press
[ENTER]. Then the calculator will want a “right bound”, so give it a number
larger than 8, say a 9 or a 10 and press [ENTER]. Then the calculator says
“Guess?”, so press [ENTER] again, and the following screen appears:
This means that the zero or
root is at x = 8. Don’t worry about the fact that it says y = 230. You are
working with an exponential graph, which is UNBELIEVABLY steep. On this
calculator one pixel (that is one DOT!) of movement to the right moves about
.21 units to the right or left, but it moves a vertical distance of
somewhere around 4.3 x1013 up or down. This is close as the
calculator can come to y = 0.
Final Answer: x= 8
P. 523 # 16.
Solution:
Take the ln of both sides of
the equation:
Using the second law of
logarithms:
Using the distributive
property:
Get all x terms on the left
side by subtracting 
from
each side of the equation:
Get the x
variable in one place by factoring out the common factor of x:
Solve for x by dividing each
side of the equation by 
In the final step, parentheses
are included that would be suitable for calculating the answer with a TI 83
or TI 84. To calculate with a TI 85 or TI 86, use parentheses as indicated
in the previous step.
Calculate with your calculator:
x ≈
-4.64385618978 . . . or -4.64
You may also want to use the
graphing calculator to obtain the approximate solution to this exercise by
setting the equation equal to zero, and graphing
. Never mind what the graph looks like, just use
the “zeros” function [2nd] [F4 (Calc)] [2 (zeros)] of the TI
83 or TI 84 or the “ROOT” method of the TI 85 or TI 86 to find the
x-intercept of that graph. It will amazingly enough be exactly the same (at
least to calculator accuracy) as the answer given above! Please see
the graph below that was contributed by Department Chairman
Lane Vosbury.
:
P. 523 # 17.
Solution:
Take the
LN
of both sides of the equation:
Using the
second law of logarithms:
Using the
distributive property:
Get all x
terms on the left side by subtracting
from
each side of the equation and get the non-x terms on the right side by
adding 
:
Get the
x variable in one place by factoring out the common factor of
x:
Solve for
x by dividing each side of the equation by
In the
final step, parentheses are included that would be suitable for calculating
the answer with a TI 83 or TI 84. To calculate with a TI 85 or TI 86, use
parentheses as indicated in the previous step.
Calculate
with your calculator:
x ≈ −2.451964916 . . .
which rounds off to
−2.45
P. 523 # 19.
Solution:
Take the
LN
of both sides of the equation:
Using the
second law of logarithms:
Using the
distributive property:
Get all x
terms on the left side by subtracting
from
each side of the equation and get the non-x terms on the right side by
adding 
:
Get the
x variable in one place by factoring out the common factor of
x:
Solve for
x by dividing each side of the equation by
In the
final step, parentheses are included that would be suitable for calculating
the answer with a TI 83 or TI 84. To calculate with a TI 85 or TI 86, use
parentheses as indicated in the previous step.
Calculate
with your calculator:
Note: It
came out even!!! Could this be a “contrived” problem?? Could it be solved
as in the first few problems in this section using the laws of exponents??
Hmmm . . .
P. 525 # 23.
Solution:
Since all the logarithms are
already on the same side of the equation, you should begin by combining the
logarithm terms on the left side into a single logarithm, using the first
law of logarithms. Remember, when you have a sum of logarithms, you get the
log of a product.
Now
convert this logarithmic form to exponential form, using the basic
definition of logarithms.
means
the same as 
Solve as a
quadratic equation.
This
factors into:
Of these
answers, 
must
be rejected, since it results in a log of a negative in the original
problem. The final answer is 
.
Extra Problem from algebra.com.
Solution:
Since all the logarithms are already on the same side of the equation, you
should begin by combining the logarithm terms on the left side into a single
logarithm, using the first law of logarithms. Remember, when you have a sum
of logarithms, you get the log of a product.
Now convert this logarithmic
form to exponential form, using the basic definition of logarithms.
means
the same as 
Solve as a quadratic equation.
This factors into:
Of these answers,
must
be rejected, since it results in a log of a negative in the original
problem. The final answer is 
.
Also it checks! You can
substitute into
the original equation:
P. 526 # 27. 
Solution:
You should begin by getting all the logarithm
terms on one side, and any non-log terms on the other side. In this case,
subtract 
from
each side of the equation. This leaves:
By the second law of
logarithms, the difference of two logarithms can be expressed as a quotient:
Now convert this
logarithmic form to exponential form, using the basic definition of
logarithms.
means
the same as 
Cross multiply, or
multiply both sides of the equation by the LCD—it gives you the same result:
Solve as any linear
equation.
This answer is
acceptable, since when substituted into the original equation, it always
results in the logs of positive numbers.
P. 527 # 29.
Solution:
You should begin by getting all
the logarithm terms on one side, and any non-log terms on the other side.
In this case, subtract 
from each side of the equation. This leaves:
By the second law of
logarithms, the difference of two logarithms can be expressed as a quotient:
Now convert this logarithmic
form to exponential form, using the basic definition of logarithms.
means the same as
Cross multiply, or multiply
both sides of the equation by the LCD—it gives you the same result:
Solve as any linear equation.
P. 528 # 33. 
Solution:
In the previous example, you
were instructed to get all the log terms on one side and the non-log terms
on the other side. However, in this case, there are no non-log terms, so
there is no need to do this step. It is a good idea in this case to use the
second law of logarithms to make the log of a quotient on the
left. 
Next,
remember the “This
Equals That” Theorem
that was mentioned in #3? Well,
“This Equals That”
applies to logarithms as well as exponents:
“This Equals
That Theorem”:
If ,
then 
Now, you
may either cross multiply, or multiply both sides of the equation by the
LCD—it gives you the same result:
Solve as a
quadratic equation.
Set the
equation equal to zero:
Factor the
trinomial:
Set each
factor equal to zero:
Now, you
must check these answers by substituting them back into the original
equation to make sure that neither answer gives you a log of a negative. As
it turns out, BOTH of these answers result in logs of negatives, so neither
answer is acceptable. Both must be rejected! Therefore the final answer is
NO SOLUTION!
P. 528 # 34.
Solution:
As in the previous example, there are no non-log terms in the equation, so
it will not be necessary to get all the log terms on one side. It is a
good idea in this case to use the second law of logarithms to make the log
of a quotient
on the left and also on the right side of the equation.
Next, remember the
“This Equals
That” Theorem
that was mentioned in the
previous exercise? Well,
“This Equals That”
is back again!!
“This Equals
That Theorem”:
If
, then 
Now, you may either cross
multiply, or multiply both sides of the equation by the LCD—it gives you the
same result:
Solve as a quadratic equation.
Set the equation equal to zero:
Factor the trinomial:
Set each factor equal to zero:
Both answers are acceptable,
since neither answer results in a log of a negative.
P. 528 # 35. 
Solution:
As in the previous
examples, there are no non-log terms, so there is no need to get all the
logs on one side of the equation. It is a good idea in this case to use the
first law of logarithms to make the log of a
product
on the left and also on the right side of the equation.
Next,
remember the “This
Equals That” Theorem
from the previous exercise? Well, here it is again:
“This
Equals That Theorem”:
If
,
then 
Solve as a
quadratic equation, and set equal to zero.
Factor the
trinomial:
Set each
factor equal to zero:
The first
answer
,
is acceptable, but you must reject the
since
it results in a log of a negative.
P. 528 # 36.
Solution:
As in
the previous examples, there are no non-log terms, so there is no need to
get all the logs on one side of the equation. It is a good idea in this
case to use the first law of logarithms to make the log of a
product
on the left and also on the right side of the
equation.
Next, remember the “This Equals That”
Theorem from
the previous exercises? Well, here it is again:
“This
Equals That Theorem”:
If
,
then 
Solve as a
quadratic equation, and set equal to zero.
Factor the
trinomial:
Set each
factor equal to zero:
The first
answer
must
rejected since it results in a log of a negative (actually it results in two
negatives, as it most always does!). However, second answer
is
acceptable.
Extra Problem from Keith.
Find the exact value, and also
calculate the answer to the nearest millionth.
Solution:
This really means
which
means 
Now solve for x:
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