Exponential and logarithmic equations

EXPONENTIAL AND LOGARITHMIC EQUATIONSSection 3.4

EXPONENTIAL & LOG EQUATIONS In the previous sections, we covered:

a) Definitions of logs and exponential functionsb) Graphs of logs and exponential functionsc) 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 31

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


42)3(2x



x2 2log 14log2

x2 log

14 log ...807.3

142x


16)4(e2x



2xeln 4ln 2x 4ln

24ln

4e2x

x 693.0

EXPONENTIAL & LOG EQUATIONS Solve the following equations:

a)

b)

c) 148)5(e 2x

10)12(3x

605ex

2 -522ln 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


0232 xx ee

Start by rewriting the equation in quadratic form.

023)( 2 xx eeFactor the quadratic equation:

xe let x 0232 xx 0)1)(2( xx

0)1)(2( xx ee

EXPONENTIAL & LOG EQUATIONS

0)1)(2( xx ee

02 xe 01xe2xe 1xe

2lnln xe 0x

2lnx

693.x


0202 xx ee

020)( 2 xx ee

05 xe

0)4)(5( xx ee

04 xe

5xe 4xe5lnx 4lnx

609.1x errorx

EXPONENTIAL AND LOGARITHMIC EQUATIONSSection 3.4


538 x

853 x

85log3log 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

3log35log5log23log2

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


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 xx715 xx

2x

EXPONENTIAL & LOG EQUATIONS Solve the following equation

43log2 5 x


23log5 xRewrite the expression in exponential form

x352

325

x

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


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

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

310 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

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 EQUATIONSrtPeA

tPeP 08.02 te 08.02 te 08.0ln2ln

2ln08.0 t08.02ln

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?




Documents

Exponential and logarithmic equations