1. 5-3 Logarithmic Functions

2. Logarithms have many applications inside and outside mathematics
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor

3. •To find the number of payments on a loan or the time to reach an investment goal
•To model many natural processes, particularly in living systems. We perceive loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale. We also perceive brightness of light as the logarithm of the actual light energy, and star magnitudes are measured on a logarithmic scale.
•To measure the pH or acidity of a chemical solution. The pH is the negative logarithm of the concentration of free hydrogen ions.
•To measure earthquake intensity on the Richter scale. (moment magnitude scale)
•To analyze exponential processes. Because the log function is the inverse of the exponential function, we often analyze an exponential curve by means of logarithms. Plotting a set of measured points on “log-log” or “semi-log” paper can reveal such relationships easily. Applications include cooling of a dead body, growth of bacteria, and decay of a radioactive isotopes. The spread of an epidemic in a population often follows a modified logarithmic curve called a “logistic”.
•To solve some forms of area problems in calculus. (The area under the curve 1/x, between x=1 and x=A, equals ln A.)

4. Other uses that are more aligned with the scope of this course are determining decibel levels for sound (loudness) and severity of earthquakes. Logarithms are used in other areas but the math needed to apply them is advanced beyond this class. We will use a few of these concepts to learn how to deal with logs through that use hopefully you learn an appreciation for the need of logs.

5. In its simplest form, a logarithm answers the question:
First of all, what is a logarithm? You’ve worked with them before, you’ve learned the rules before, but do you really know what one is?
In its simplest form, a logarithm answers the question:
How many of one number do we multiply to get another number?
For example how many 2s do you need to multiply together to get 8?
The log then would be 3.
The notation would be
How many of the bases “2” do you need to multiply by itself to get 8?

6. Remember logs and exponentials are inverses of one another.
Take your calculator, find the log button, the log button on your calculator uses a base of 10. Later we can change that, but for right now our calculator is only useful when the base is 10.
Verify the above value with your calculator. What does this mean.
To get a value of 6.3, you need to multiply 10 by itself 0.8 times.

7. Common Log
A log that involves a base of 10 is referred to as the common log. The definition is as follows ….

8. General Log Rule for logs of bases other than 10

9. Convert these logs into exponential form

10. Write the following in log form
103 = 1, = = = 5 70 = = 1/ /2 = 5

11. KEY Properties of logs for quick substitution.

12. Laws of Logs
You need to know these and now to use these so that in the coming days we can solve equations that involve logs.

13. Write in expanded form

14. Write in condensed form as one logarithm

15. Write as one logarithm

16. Use the laws of logs to evaluate expressions

17. Natural Log “ln”
We can take a log that has a base of e. It would look like the following
However this is used so often that it gets its own notation
This is read as the natural log of x.

18. We use the same laws for natural log. As well as the same properties
Most of the time
when I see ln, I rewrite it as loge
Makes it easier for my thought patterns

19. Solving Logarithmic equations
In order to solve logarithmic equations the bases must be consistent throughout the entire equation.
If the equation has parts that needs to be condensed through the use of the laws of logarithms you must do that first. Ultimately we want one of two situations to arise, either have 0 or 1 logs on each side of the equation with the same base.

20. Example
This is very generic, but if both logs have the same bases then we can ignore the log part and set x=5. Thus arriving at our solution.
Obviously x and 5 can be more complex expressions.

22. We can also have equations where we must first condense one or both sides of the equation.

23. Need to re-write so that e shows up

25. Chapter 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions
5.3 Logarithms and Their Properties
5.4 Logarithmic Functions
5.5 Exponential and Logarithmic Equations and Inequalities
5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

26. 5.3 Logarithms and Their Properties
For all positive numbers a, where a  1,
A logarithm is an exponent, and loga x is the exponent to which a must be raised in order to obtain x. The number a is called the base of the logarithm, and x is called the argument of the expression loga x. The value of x must always be positive.

27. 5.3 Examples of Logarithms
Exponential Form Logarithmic Form
Example Solve
Solution

28. 5.3 Solving Logarithmic Equations
Example Solve a)
Solution
Since the base must
be positive, x = 2.

29. 5.3 The Common Logarithm – Base 10
Example Evaluate
Solution Use a calculator.
For all positive numbers x,

30. 5.3 Application of the Common Logarithm
Example In chemistry, the pH of a solution is defined as
where [H3O+] is the hydronium ion
concentration in moles per liter. The pH value is a measure of
acidity or alkalinity of a solution. Pure water has a pH of 7.0,
substances with pH values greater than 7.0 are alkaline, and
substances with pH values less than 7.0 are acidic.
Find the pH of a solution with [H3O+] = 2.5×10-4.
Find the hydronium ion concentration of a solution with pH = 7.1.
Solution
pH = –log [H3O+] = –log [2.5×10-4]  3.6
7.1 = –log [H3O+]  –7.1 = log [H3O+]
 [H3O+] =  7.9 ×10-8

31. 5.3 The Natural Logarithm – Base e
On the calculator, the natural logarithm key is usually found in conjunction with the e x key.
For all positive numbers x,

32. 5.3 Evaluating Natural Logarithms
Example Evaluate each expression.
Solution

33. 5.3 Using Natural Logarithms to Solve a Continuous Compounding Problem
Example Suppose that $1000 is invested at 3%
annual interest compounded continuously. How
long will it take for the amount to grow to $1500?
Analytic Solution

34. 5.3 Using Natural Logarithms to Solve a Continuous Compounding Problem
Graphing Calculator Solution
Let Y1 = 1000e0.03t and Y2 = 1500.
The table shows that when time (X) is 13.5 years, the
amount (Y1) is  1500.

35. 5.3 Properties of Logarithms
Property 1 is true because a0 = 1 for any value of a.
Property 2 is true since in exponential form:
Property 3 is true since logak is the exponent to which a must be raised in order to obtain k.
For a > 0, a  1, and any real number k,
loga 1 = 0,
loga ak = k, 3.

36. 5.3 Additional Properties of Logarithms
For x > 0, y > 0, a > 0, a  1, and any real number r,
Product Rule
Quotient Rule
Power Rule

37. 5.3 Additional Properties of Logarithms
Examples Assume all variables are positive.
Rewrite each expression using the properties of
logarithms.

38. 5.3 Example Using Logarithm Properties
Example Assume all variables are positive. Use the
properties of logarithms to rewrite the expression
Solution

39. 5.3 Example Using Logarithm Properties
Example Use the properties of logarithms to write
as a single logarithm
with coefficient 1.
Solution

40. 5.3 The Change-of-Base Rule
Proof Let
Change-of-Base Rule
For any positive real numbers x, a, and b, where
a  1 and b  1,

41. 5.3 Using the Change-of-Base Rule
Example Evaluate each expression and round to
four decimal places.
Solution Note in the figures below that using
either natural or common logarithms produce the
same results.

42. 5.3 Modeling the Diversity of Species
Example One measure of the diversity of species
in an ecological community is the index of diversity
where P1, P2, , Pn are the proportions of a sample
belonging to each of n species found in the sample.
Find the index of diversity in a community with two
species, one with 90 members and the other with 10.

43. 5.3 Modeling the Diversity of Species
Solution Since there are a total of 100 members in
the community, P1 = 90/100 = 0.9, and
P2 = 10/100 = 0.1.
Interpretation of this index varies. If two species are
equally distributed, the measure of diversity is 1. If
there is little diversity, H is close to 0. In this case
H  0.5, so there is neither great nor little diversity.

44. More Examples

49. Solve Exponential Equations using logs (when variable is in exponent)