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Chapter 3
3-4 solving exponential and logarithmic functions
SAT Problem of the day
objectives
 Solve simple and more complex exponential and
logarithmic functions
Exponential and logarithmic
functions
 Remember that exponential and logarithmic functions are one-toone functions. That means that they have inverses. Also recall that
when inverses are composed with each other, they inverse out and
only the argument is returned. We're going to use that to our
benefit to help solve logarithmic and exponential equations.
 Please recall the following facts:
 loga ax = x
 log 10x = x
 ln ex = x
 a loga x = x
 10log x = x
 eln x = x
Solving exponential equations
 Isolate the exponential expression on one side.
 Take the logarithm of both sides. The base for the logarithm
should be the same as the base in the exponential expression.
Alternatively, if you are only interested in a decimal
approximation, you may take the natural log or common log of
both sides (in effect using the change of base formula)
 Solve for the variable.
 Check your answer. It may be possible to get answers which
don't check. Usually, the answer will involve complex numbers
when this happens, because the domain of an exponential
function is all reals.
Example#1
Solve the equation 3 x=27.
If x = 1, then 3 1 = 3.
If x = 2, then 3 2 = 9.
If x = 3, then 3 3 = 27.
Since 3 3 = 27, x = 3 is
the solution to the equation 3 x = 27.
Solve this by trying some small values
of x
(i.e. try x = 1, 2, etc. )
Summarize the solution.
Example#2
Solve the equation 3 e 2x = 35 for x and use a calculator to give a 4 decimal place approximation ans
e2x = 35 / 3 .
Divide both sides by 3 in order to
prepare this equation
for the application of the natural log
function.
ln e2x = ln (35 / 3 ).
Apply the natural log function to both
sides.
2x = ln ( 35 / 3 )
x =(1/2) ln (35 / 3 )
1.2284
Use the inverse property of logs and
exponents.
Solve for x.
Use a calculator to find a 4 decimal
approximation
to the answer.
Example#3
Solve the equation 3 + e 3x + 2= 23 for x and use a calculator to give a 4 decimal place a
e 3x + 2= 20
Subtract 3 from both sides of the
equation in order to prepare it
for the application of the natural log
function.
ln e 3x + 2= ln (20).
Apply the natural log function to both
sides.
3x + 2 = ln ( 20 )
x =(1/3) ( -2 + ln 20 )
.3319
Use the inverse property of logs and
exponents.
Solve for x.
Use a calculator to find a 4 decimal
approximation
to the answer.
Example#4
 Solve for x : 4·52x = 64 .
Solution:
4·52x = 64
52x = 16
log552x = log516
2x = log516
2x =
2x 1.723
x 0.861
Student guided practice
 Do problems 1-6 in your worksheet
Logarithmic Equations
 Use properties of logarithms to combine the sum, difference,
and/or constant multiples of logarithms into a single logarithm.
 Apply an exponential function to both sides. The base used in
the exponential function should be the same as the base in the
logarithmic function. Another way of performing this task is to
write the logarithmic equation in exponential form.
 Solve for the variable.
 Check your answer. It may be possible to introduce extraneous
solutions. Make sure that when you plug your answer back into
the arguments of the logarithms in the original equation, that
the arguments are all positive. Remember, you can only take the
log of a positive number.
Example#5
 Solve for x : log3(3x) + log3(x - 2) = 2 .
log3(3x) + log3(x - 2) = 2
log3(3x(x - 2)) = 2
32 = 3x(x - 2)
9 = 3x 2 - 6x
3x 2 - 6x - 9 = 0
3(x 2 - 2x - 3) = 0
3(x - 3)(x + 1) = 0
x = 3, - 1
Example#5
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ln (x + 4) + ln (x - 2) = ln 7
First we use property 1 of logarithms to combine the terms on the left.
ln (x + 4)(x - 2) = ln 7
Now apply the exponential function to both sides.
eln (x + 4)(x - 2) = eln 7
The logarithmic identity 2 allows us to simplify both sides.
(x + 4)(x - 2) = 7
x2 + 2x - 8 = 7
x2 + 2x - 15 = 0
(x - 3)(x + 5) = 0
x = 3 or x = -5
x = 3 checks, for ln 7 + ln 1 = ln 7.
Example#6
 solve −6log 3 (x − 3) = −24 for x
Example#7
 log 9 (x + 6) − log 9x = log 92
Do problems
 Do problems 1-8 in your worksheet
Video
 Let’s watch solving exponential and logarithmic
function
Homework
 Do problems 35 -42 in your book page 217
closure
 Today we learned to solve exponential and
logarithmic functions
 Next class we are going to learn about exponential
and logarithmic models
Have a nice day