Solving Logarithmic Equations

Math 1111Tosha Lamar, Georgia Perimeter College Online

Who would ever need to solve a logarithmic equation?

There are several areas in which logarithmic equations must be used such as

An archeologist wants to know how old a fossil is using the measure of the Carbon-14 remaining in a dinosaur bone.

A meteorologist needs to measure the atmospheric pressure at a given altitude.

A scientist needs to know how long it will take for a certain liquid to cool from 75 degrees to 50 degrees.

We need to measure how loud (in decibels) a rock concert is to determine if it could cause long-term hearing loss.

. . . just to name a few

When we finish this lesson we will be able to solve a problem like this:

In 1989, an earthquake that measures 7.1 on the Richter scale occurred in San Fransico, CA. Find the amount of energy, E , released by this earthquake.

On the Richter scale, the magnitude, M, of an earthquake depends on the amount of energy, E, released by the earthquake according to this formula. M is a number between 1 and 9. A destructive earthquake usually measures greater than 6 on the Richter scale.

Illustration from IRIS Consortium

The amount of energy, measured in ergs, is based on the amount of ground motion recorded by a seismograph at a known distance from the epicenter of the earthquake.

After we have practiced solving some logarithmic equations we will go back and find the answer to this problem!

1995 Tokyo Earthquake

From our previous lessons, recall the properties of logarithms.We will refer back to these throughout the lesson.

Properties of Logarithms

Definition of Logarithm For x > 0 and b > 0, b 1y = logb x is equivalent to by = x

Product Property Let b, M, and N be positive real numbers with b 1.logb (MN) = logb M + logb N

Quotient Property Let b, M, and N be positive real numbers with b 1.

Power Property Let b and M be positive real numbers with b 1, and let p be any real number.logb Mp = p logb M

NMN

Mbbb logloglog

Properties of Logarithms (cont.)

Exponential-Logarithmic Inverse Properties

For b > 0 and b 1, logb bx = x and

One-to-One Property of Exponents

If bx = by then x = y

One-to-One Property of Logarithms

If logb M = logb N, then M = N(M > 0 and N > 0)

Change-of-Base Formula For any logarithmic bases a and b, and any positive number M,

b

MM

a

ab log

loglog

xb xb log

log4 x = 5

45 = x

x = 1024

Check your answer!

log4 x = 5

log4 1024

ln x = 3Round your answer to the nearest hundredth.

The “base” of a natural logarithm (ln) is e

e3 = x

x = 20.09(rounded to the nearest hundredth)

Check your answer!

ln x = 3

ln 20.09 It is not “exactly” 3 since you rounded the answer to the nearest hundedth.

log5 (x-6) = -3 Write your answer as a fraction in lowest terms.

5-3 = x – 6

= x – 6

Check your answer!

log5 (x-6) = -3

log5 ( - 6) 125

751

log5 ( ) 125

1

Correct!!

5 ln (3x) = 10Write your answer as a decimal rounded to the nearest hundredth.

Divide both sides of the equation by 5

ln (3x) = 2

e2 = 3x

The “base” of a natural logarithm (ln) is e

x = 2.46

2log3 (x-1) = 4 – log3 5Write your answer as a decimal rounded to the nearest hundredth.

log3 (x-1)2 = 4 – log3 5

log3 (x-1)2 + log3 5 = 4

log3 5(x-1)2 = 4

log3 5(x-1)2 = 4

34 = 5(x-1)2

81 = 5(x2 – 2x + 1)

81 = 5x2 – 10x + 5

0 = 5x2 – 10x - 76

(continued from previous slide)

(square (x-1) using FOIL) and 34 = 81

Distribute the 5

Subtract 81 from both sides to get the equation equal to zero

a

acbbx

2

42

5x2 – 10x – 76 = 0 Solve using quadratic

formula

)5(2

)76)(5(4)10()10( 2 x

10

162010x

The solution set is x = {5.02, -3.02}

log2 (x – 3) + log2x – log2 (x +2) = 2

log2 x(x – 3) – log2 (x +2) = 2

2)2(

)3(log2

x

xx

2)2(

)3(log2

x

xx(continued from previous slide)

22 = 4

Multiply both sides by (x+2)

)2(

)3(22

x

xx

)2(

)3(4

x

xx

)3()2(4 xxx

xxx 384 2

870 2 xx

Distribute the 4

Subtract 4x and subtract 8 to get one side of the equation equal to zero

870 2 xx

)1)(8(0 xx

0108 xx

x = 8

x = -1

Solution set: x = {8, -1}

Set each factor equal to zero (zero product property)

3 log x = log 64

log x3 = log 64

x3 = 64

33 3 64xx = 4

2 log x – log 6 = log 96

log x2 – log 6 = log 96

96log6

log2

x

96log6

log2

x

966

2

x

5762 x

24xWhy not also -24??

Look at the original equation – if we allowed x to be -24 we would be taking the log of a negative number!

Multiply both sides by 6

log(x – 3) + log 4 = log 108

log 4(x – 3) = log 108

log (4x – 12) = log 108

4x – 12 = 108

4x = 120 x = 30

log2(x + 12) – log2 (x + 3) = log2 (x - 4)

)4(log3

12log 22

xx

x

43

12

xx

x

)3)(4(3

123

xxx

xx

1212 2 xxx

2420 2 xx

)4)(6(0 xx

606 xx

404 xx

(multiply both sides by (x+3)

Multiply (x-4)(x+3) using FOIL

Subtract x and subtract 12 from both sides to get the equation equal to zero

Factor

Set each factor equal to zero

Always check your answers

log2(6 + 12) – log2 (6 + 3) = log2 (6 - 4)

log2(18) – log2 (9) = log2 (2)

Check x = -4

CANNOT TAKE LOG OF NEGATIVE NUMBER !

log2(-4 + 12) – log2 (-4 + 3) = log2 (-4 - 4)

log2(8) – log2 (-1) = log2 (-8)

Check x = 6

log2(2)= log2 (2)

FINAL ANSWER: x = 6

Now we can go back to our earthquake problem!

In 1989, an earthquake that measures 7.1 on the Richter scale occurred in San Fransico, CA. Find the amount of energy, E , released by this earthquake.

)10log(log3

21.7 8.11 E

)10log8.11(log3

21.7 E

)8.11(log3

21.7 E

)8.11(log3

2

2

31.7

2

3 E

8.11log65.10 E

Elog45.22

(because log 10 = 1)

Multiply by (3/2) to get rid of the fraction on the right

(3/2) * 7.1 = 10.65

Add 11.8 to both sides

E45.2210

Therefore, the amount of energy, E , released by this earthquake was 2.82 x 1022

ergs.