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
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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
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 will always be positive.
Exponential Form Logarithmic Form
Since the base must
be positive, x = 2.
For all positive numbers x,
SolutionUse a calculator.
ExampleIn 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 a pH greater than 7.0 are alkaline, and those
less than 7.0 are acidic.
[H3O+] = 10-7.1 7.9 ×10-8
For all positive numbers x,
Example Suppose that $1000 is invested at 3%
annual interest, compounded continuously. How
long will it take for the amount to grow to $1500?
Graphing Calculator Solution
Let Y1 = 1000e.03t and Y2 = 1500.
The table shows that when time (X) is 13.5 years, the
amount (Y1) is 1499.3 1500.
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 x > 0, y > 0, a > 0, a 1, and any real number r,
Examples Assume all variables are positive. Rewrite each
expression using the properties of logarithms.
Example Assume all variables are positive. Use the
properties of logarithms to rewrite the expression
Example Use the properties of logarithms to write
as a single logarithm
with coefficient 1.
For any positive real numbers x, a, and b, where
a 1 and b 1,
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
Example One measure of the diversity of species
in an ecological community is the index of diversity,
and 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 where
there are two species, with 90 of one species and 10
of the other.
Solution Since there are a total of 100 members in
the community, P1 = 90/100 = .9, and P2 = 10/100 = .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 .5, so there is neither great nor little diversity.