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

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.

5.3 Examples of Logarithms Exponential Form Logarithmic Form Example Solve Solution

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

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

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+] = 10-7.1  7.9 ×10-8

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,

5.3 Evaluating Natural Logarithms Example Evaluate each expression. Solution

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

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 1499.3  1500.

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.

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

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

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

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

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,

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.

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.

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.