2. 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

3. 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.

4. 5.3 Examples of Logarithms
Exponential Form Logarithmic Form
Example Solve
Solution

5. 5.3 Solving Logarithmic Equations
Example Solve a)
Solution
Since the base must
be positive, x = 2.

6. 5.3 The Common Logarithm – Base 10
Example Evaluate
Solution Use a calculator.
For all positive numbers x,

7. 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

8. 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,

9. 5.3 Evaluating Natural Logarithms
Example Evaluate each expression.
Solution

10. 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

11. 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.

12. 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.

13. 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

14. 5.3 Additional Properties of Logarithms
Examples Assume all variables are positive.
Rewrite each expression using the properties of
logarithms.

15. 5.3 Example Using Logarithm Properties
Example Assume all variables are positive. Use the
properties of logarithms to rewrite the expression
Solution

16. 5.3 Example Using Logarithm Properties
Example Use the properties of logarithms to write
as a single logarithm
with coefficient 1.
Solution

17. 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,

18. 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.

19. 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.

20. 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.