Objectives: 1. Solve exponential and logarithmic equations. 2.
Solve a variety of application problems by using exponential and
logarithmic equations. Solving Exponential and Logarithmic
Equations 5.6
Slide 2
Methods of Solving Exponential Equations 1. Rewrite the bases
of the powers on both sides so they are the same. This method is
the fastest to use but only works for certain situations. It uses a
fact that we often take for granted. For instance, in the equation
2 x = 2 3, we would say that x = 3. If the bases are equal, then
the exponents must be equal as well. If the equation were modified
slightly, say 2 x 1 = 2 3, we would still set the exponents equal,
but now we would say that x 1 = 3, therefore x = 4. 2. Take a
logarithm of both sides. This methods always works, and for simple
situations it is rather quick, but it can also be very cumbersome.
Especially if variables are on both sides of the equation.
Slide 3
Example #1 Powers of the Same Base Solve the equation and
confirm your solution with a graph. A. Rewrite both bases to base
3.
Slide 4
Example #1 Powers of the Same Base Solve the equation and
confirm your solution with a graph. B. Rewrite to base 2, remember
fractions have negative exponents.
Slide 5
Example #1 continued Powers of the Same Base Solve the equation
and confirm your solution with a graph. C. Rewrite to base 5.
Slide 6
Example #1 continued Powers of the Same Base Solve the equation
and confirm your solution with a graph. D. This time the graphs
with the intersection method were too difficult to read, so the
intercept method was used to verify the two solutions.
Slide 7
Example #2 Logarithms on Both Sides Solve the equation.
Solutions can be confirmed with a graph. Problems like this can be
confusing because students will recognize that 2(3) = 6.
Unfortunately, that will not work because we have to be able to
rewrite both sides to the same base of a power for the first method
to be used. Our only choice then is the second method. Notice how
common logs and natural logs both work the same way. Choosing which
one to use basically comes down to personal preference.
Slide 8
Example #3 Solve the equation. This type can be very tricky:
1.First take a logarithm of both sides (common or natural). 2.Swing
the exponents in front using the properties of logarithms.
3.Distribute. 4.Rewrite the expression so that x terms are on one
side of the equation and everything else is on the other side.
Remember if you change the side, change the sign. 5.Factor out the
GCF of x. 6.Divide.
Slide 9
Example #3 Solve the equation. Typing the answer in the
calculator can be difficult: Alternatively you can first simplify
the answer further by using the properties of logarithms.
Slide 10
Example #4 Solve the equation. 1.Multiply both sides by e x.
This will remove the negative exponent on the e. 2.Rewrite the
expression equal to 0. 3.Substitute u = e x.
Slide 11
Example #4 Solve the equation. 1.Use the quadratic formula to
solve for u. 2.Substitute e x back in for u. 3.Take the natural
logarithm of both sides to eliminate the e on the left side. 4.Only
the positive solution from the quadratic formula is used because
you cant take the log of a negative number.
Slide 12
Solving Logarithmic Equations To solve a logarithmic equation,
it is often necessary to condense the expression down to a single
logarithm on each side first. This means all the properties of
logarithms will come into play. Additionally, constant terms and
logarithms cannot be on the same side. After an individual
logarithm is created on each side of the equation, the logarithm
can be removed using its inverse, a power of the same base.
Finally, make sure to check for extraneous solutions since the
domain of logarithms are limited to numbers greater than 0.
Slide 13
Example #5 Solve the equation and confirm your solution with a
graph. First note that 2(ln x) is the same thing as 2ln x, which
had its coefficient placed up top as an exponent. Secondly, e was
chosen as the base of the power because it is the inverse of the
natural log. After the logarithms are removed and the expression is
set equal to 0, we get a quadratic that cannot factor and will
require the Quadratic Formula.
Slide 14
Example #5 Solve the equation and confirm your solution with a
graph. Take for instance the expression: ln(x 5) If the solutions
to an equation with that expression were 4 and 5, even though both
answers are positive, when plugged back in we get: ln(4 5) = ln( 1)
ln(5 5) = ln(0) Both of these expressions are undefined. When
solving this problem we get two solutions. It is important to check
both solutions because any value for x (positive or negative) that
makes the resulting logarithmic expression negative or 0 will
result in no solution.
Slide 15
Example #5 Solve the equation and confirm your solution with a
graph. Therefore, the only valid solution is the positive solution:
Do not assume this is always the case. A more efficient way of
checking will be with the graphing calculator.
Slide 16
Example #5 Solve the equation and confirm your solution with a
graph. Note: You will have to move the cursor to select each curve
when using the intersect method. Secondly, the intersection isnt
showing on this screen, but it is there. A better view:
Slide 17
Example #6 Solve the equation. The trick to this problem is
recognizing to the change the addition of two of the same
expressions into multiplication by 2. Then both sides are divided
by 2 and the problem is solved fairly straightforward.
Slide 18
Example #6 Solve the equation. Here a second method of solving
the same problem is shown but this time two possible answers
arrive. After checking for extraneous solutions the same solution
is identified.
Slide 19
Example #7 Solve the equation. The only valid solution is x =
3.
Slide 20
Example #8 Radioactive Decay It is determined that a mummy has
lost 32% of its original carbon-14. When did the person die? We
also need to know the half-life of carbon-14 which is 5730 years
old. Previously, this type of problem was solved by graphing. Now
we will solve it algebraically. If 32% of carbon-14 is lost, then
100% 32% = 68%
Slide 21
Example #8 Radioactive Decay It is determined that a mummy has
lost 32% of its original carbon-14. When did the person die? We
take the logarithm of both sides. Then we swing the exponent out
front of the logarithm, and solve for x.
Slide 22
Example #9 Compound Interest If $8000 is to be invested at 6%
per year, compounded monthly, in how many years will the investment
be worth $22,520? Make sure to isolate the base of 1.005 before
taking the log.
Slide 23
Example #10 Population Growth The population of certain species
of rabbit increases exponentially. At the beginning of a 12-month
study, there were 50 rabbits, and at the end of the study there
were 164 rabbits. Find the following: A. Find the rate of growth.
After solving for a, dont forget to substitute 1 + r back in for a
to find r. Here we see the growth rate is about 10%.
Slide 24
Example #10 Population Growth The population of certain species
of rabbit increases exponentially. At the beginning of a 12-month
study, there were 50 rabbits, and at the end of the study there
were 164 rabbits. Find the following: B. Write the function for
this population. To keep the function more accurate, we will need
to use a growth rate more accurate than 10%. Alternatively, we can
use the value for a we found earlier.
Slide 25
Example #10 Population Growth The population of certain species
of rabbit increases exponentially. At the beginning of a 12-month
study, there were 50 rabbits, and at the end of the study there
were 164 rabbits. Find the following: C. When will the population
reach 1000?
Slide 26
Example #11 Population Growth The population of a certain
bacteria culture at time t hours is given by the function: How long
will it take for the bacteria population to reach 8000? The
equation we will need to solve is given by:
Slide 27
Example #11 Population Growth How long will it take for the
bacteria population to reach 8000?
