E XPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4.
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Transcript of E XPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4.
EXPONENTIAL & LOG EQUATIONS
In the previous sections, we covered:
a) Definitions of logs and exponential functions
b) Graphs of logs and exponential functions
c) Properties of logs and exponential functions
In this section, , we are going to study procedures for solving equations involving logs and exponential equations
0 36 9e e :e.g. x2x
EXPONENTIAL & LOG EQUATIONS
In the last section, we covered two basic properties, which will be key in solving exponential and log equations.
1. One-to-One Properties
2. Inverse Properties
a) yx aa ylog x log b) aa
xaalog a) xlog b) aa
y x y x
x x
EXPONENTIAL & LOG EQUATIONS
We can use these properties to solve simple equations:
32 2 x 52 2 x 5 x
9 3
1
x23 3 x 2- x
3 ln x 3ln ee x -3e x
EXPONENTIAL & LOG EQUATIONS
When solving exponential equations, there are two general keys to getting the right answer:
1. Isolate the exponential expression
2. Use the 2nd one-to-one property
ylog x log aa y x
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
724x
Isolate the exponential expression:
Apply the 2nd one-to-one property
x4 4log 72log4
x4 log
72 log ...085.3
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
42)3(2x
Isolate the exponential expression:
Apply the 2nd one-to-one property
x2 2log 14log2
x2 log
14 log ...807.3
142x
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
16)4(e2x
Isolate the exponential expression:
Apply the 2nd one-to-one property
2xeln 4ln 2x 4ln
2
4ln
4e2x
x 693.0
EXPONENTIAL & LOG EQUATIONS
Solve the following equations:
a)
b)
c) 148)5(e 2x
10)12(3x
605ex
2 -5
22ln x
3ln
4ln x
55ln x 4.007 x
1.262 x
0.518- x
EXPONENTIAL & LOG EQUATIONS
Solving Equations of the Quadratic Type
Two or more exponential expressions
Similar procedure to what we have been doing
Algebra is more complicated
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
0232 xx ee
Start by rewriting the equation in quadratic form.
023)( 2 xx ee
Factor the quadratic equation:
xe let x 0232 xx 0)1)(2( xx
0)1)(2( xx ee
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
0202 xx ee
020)( 2 xx ee
05 xe
0)4)(5( xx ee
04 xe
5xe 4xe
5lnx 4lnx
609.1x errorx
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
538 x
8
53 x
8
5log3log 33 x
3log8
5log x 428.0 x
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
232 35 xx
Since these are exponential functions of a different base, start by taking the log of both sides
232 3log5log xx
3log)23(5log)2( xx
EXPONENTIAL & LOG EQUATIONS
3log)23(5log)2( xx
5log25log x 3log23log3 x
3log35log xx 5log23log2
)3log35(log x
3log35log
5log23log2
x
5log23log2
212.3
EXPONENTIAL & LOG EQUATIONS
So far, we have solved only exponential equations
Today, we are going to study solving logarithmic equations
Similar to solving exponential equations
EXPONENTIAL & LOG EQUATIONS
Just as with exponential equations, there are two basic ways to solve logarithmic equations
1) Isolate the logarithmic expression and then write the equation in equivalent exponential form
2) Get a single logarithmic expression with the same base on each side of the equation; then use the one-to-one property
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
2ln x
Isolate the log expression:
Rewrite the expression in its equivalent exponential form
xe 2
389.7x
EXPONENTIAL & LOG EQUATIONS
Solve the following equation.
0)7(log)15(log 33 xx
Get a single log expression with the same base on each side of the equation, then use the one-to-one property
)7(log)15(log 33 xx
715 xx
2x
EXPONENTIAL & LOG EQUATIONS
Solve the following equation
43log2 5 x
Isolate the log expression:
23log5 xRewrite the expression in exponential form
x352
3
25x
EXPONENTIAL & LOG EQUATIONS
In some problems, the answer you get may not be defined.
Remember, is only defined for x > 0
Therefore, if you get an answer that would give you a negative “x”, the answer is considered an extraneous solution
xy alog
EXPONENTIAL & LOG EQUATIONS
Solve the following equation
2)1(log5log 1010 xx
Isolate the log expression:
2)]1(5[log10 xx
Rewrite the expression in exponential form
)1(5102 xx
EXPONENTIAL & LOG EQUATIONS
xx 55100 2
0202 xx
0)4)(5( xx
5 ,4 x
Would either of these give us an undefined logarithm?
2)1(log5log 1010 xx
EXPONENTIAL & LOG EQUATIONS
Solve the following equations:
a)
b)
c)
5log2 x
xx ln2ln
1)3log(log xx
25
10 x 228.316 x
2 ,1 x
3
10 x
EXPONENTIAL & LOG EQUATIONS
Solve the following equation:
1)2(log)3(log 44 xx
1)2)(3(log4 xx
)2)(3(4 xx
462 xx
022 xx
EXPONENTIAL & LOG EQUATIONS
How long would it take for an investment to double if the interest were compounded continuously at 8%?
What is the formula for continuously compounding interest?
rtPeA If you want the investment to double, what would A be?
PA 2
EXPONENTIAL & LOG EQUATIONS
rtPeA
tPeP 08.02 te 08.02 te 08.0ln2ln
2ln08.0 t08.0
2ln t
It will take about 8.66 years to double.
EXPONENTIAL & LOG EQUATIONS
You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?
EXPONENTIAL & LOG EQUATIONS
You have $50,000 to invest. You need to have $350,000 to retire in thirty years. At what continuously compounded interest rate would you need to invest to reach your goal?
EXPONENTIAL & LOG EQUATIONS
For selected years from 1980 to 2000, the average salary for secondary teachers y (in thousands of dollars) for the year t can be modeled by the equation:
y = -38.8 + 23.7 ln t
Where t = 10 represents 1980. During which year did the average salary for teachers reach 2.5 times its 1980 level of $16.5 thousand?