e xponential and logarithmic equations section 3.4

EXPONENTIAL AND LOGARITHMIC EQUATIONSSection 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


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


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


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


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



42)3(2x



x2 2log 14log2

x2 log

14 log ...807.3

142x



16)4(e2x



2xeln 4ln 2x 4ln

2

4ln

4e2x

x 693.0


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


Solving Equations of the Quadratic Type

Two or more exponential expressions

Similar procedure to what we have been doing

Algebra is more complicated



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


0)1)(2( xx ee

02 xe 01xe

2xe 1xe

2lnln xe 0x

2lnx

693.x



0202 xx ee

020)( 2 xx ee

05 xe

0)4)(5( xx ee

04 xe

5xe 4xe

5lnx 4lnx

609.1x errorx

EXPONENTIAL AND LOGARITHMIC EQUATIONSSection 3.4



538 x

8

53 x

8

5log3log 33 x

3log8

5log x 428.0 x



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


3log)23(5log)2( xx

5log25log x 3log23log3 x

3log35log xx 5log23log2

)3log35(log x

3log35log

5log23log2

x

5log23log2

212.3


So far, we have solved only exponential equations

Today, we are going to study solving logarithmic equations

Similar to solving exponential 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



2ln x

Isolate the log expression:

Rewrite the expression in its equivalent exponential form

xe 2

389.7x


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


Solve the following equation

43log2 5 x


23log5 xRewrite the expression in exponential form

x352

3

25x


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


Solve the following equation

2)1(log5log 1010 xx


2)]1(5[log10 xx

Rewrite the expression in exponential form

)1(5102 xx


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


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 AND LOGARITHMIC EQUATIONSSection 3.4 - Applications



9loglog2 55 x

9loglog 52

5 x

9x2

3x 3x



1)2(log)3(log 44 xx

1)2)(3(log4 xx

)2)(3(4 xx

462 xx

022 xx


022 xx

022 xx


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


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.


You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?


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?


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?

e xponential and logarithmic equations section 3.4

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e xponential

og e quations slide

exponential equations

following equations

complicated slide

nd onetoone property

log equations

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