1. Solving Logarithmic Equations
Math 1111
Tosha Lamar, Georgia Perimeter College Online

2. 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

3. 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

4. 1995 Tokyo Earthquake
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!

5. Properties of Logarithms
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  1
y = 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
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

6. logb bx = x and 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,

7. log4 x = 5
45 = x
x = 1024

8. Check your answer!
log4 x = 5
log4 1024

9. e3 = x x = 20.09 ln x = 3 The “base” of a natural logarithm (ln) is e
Round 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)

10. Check your answer! ln x = 3 ln 20.09
It is not “exactly” 3 since you rounded the answer to the nearest hundedth.

11. log5 (x-6) = -3
Write your answer as a fraction in lowest terms.
5-3 = x – 6
= x – 6

12. Check your answer!
log5 (x-6) = -3
log5 ( )
log5 ( )
Correct!!

13. 5 ln (3x) = 10
Write your answer as a decimal rounded to the nearest hundredth.
Divide both sides of the equation by 5
ln (3x) = 2
The “base” of a natural logarithm (ln) is e
e2 = 3x
x = 2.46

14. 2log3 (x-1) = 4 – log3 5 log3 (x-1)2 = 4 – log3 5
Write 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

15. log3 5(x-1)2 = 4 34 = 5(x-1)2 81 = 5(x2 – 2x + 1) 81 = 5x2 – 10x + 5
(continued from previous slide)
34 = 5(x-1)2
81 = 5(x2 – 2x + 1)
(square (x-1) using FOIL) and 34 = 81
81 = 5x2 – 10x + 5
Distribute the 5
0 = 5x2 – 10x - 76
Subtract 81 from both sides to get the equation equal to zero

16. The solution set is x = {5.02, -3.02}
5x2 – 10x – 76 = 0
Solve using quadratic formula
The solution set is x = {5.02, -3.02}

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

18. (continued from previous slide)
22 = 4
Multiply both sides by (x+2)
Distribute the 4
Subtract 4x and subtract 8 to get one side of the equation equal to zero

19. Set each factor equal to zero (zero product property)
x = 8
x = -1
Solution set: x = {8, -1}

20. 3 log x = log 64
log x3 = log 64
x3 = 64
x = 4

21. 2 log x – log 6 = log 96
log x2 – log 6 = log 96

22. Multiply both sides by 6
Why 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!

23. log(x – 3) + log 4 = log 108 log 4(x – 3) = log 108
4x = x = 30

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

25. (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

26. log2(6 + 12) – log2 (6 + 3) = log2 (6 - 4)
Always check your answers
Check x = 6
log2(6 + 12) – log2 (6 + 3) = log2 (6 - 4)
log2(18) – log2 (9) = log2 (2)
log2(2)= log2 (2)
Check x = -4
log2( ) – log2 (-4 + 3) = log2 (-4 - 4)
log2(8) – log2 (-1) = log2 (-8)
CANNOT TAKE LOG OF NEGATIVE NUMBER !
FINAL ANSWER: x = 6

27. Now we can go back to our earthquake problem
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.

28. (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

29. Therefore, the amount of energy, E , released by this earthquake was 2
Therefore, the amount of energy, E , released by this earthquake was 2.82 x ergs.