Logarithmic Functions OBJECTIVE Convert between logarithmic and exponential equations. Solve exponential equations. Solve problems involving exponential and logarithmic functions. Differentiate functions involving natural logarithms. Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions DEFINITION: A logarithm is defined as follows: The number is the power y to which we raise a to get x. The number a is called the logarithmic base. We read as “the logarithm, base a, of x.” Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 1: Graph: First we write the equivalent exponential equation: Then, we will select some values of y to find the corresponding values of x. Next, we will plot these points, keeping in mind that x is still the first coordinate, and connect the points with a smooth curve. Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 1 (concluded): Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions THEOREM 3: Properties of Logarithms For any positive numbers M, N, and a, and any real number k: Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions THEOREM 3 (concluded): Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 2: Given find each of the following: Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 2 (continued): e ) l o g a = 1 2 = 1 2 l o g a = 1 2 Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 2 (concluded): we cannot find: Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Quick Check 1 Given and , find each of the following: a.) b.) c.) d.) e.) f.) a.) b.) c.) Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Quick Check 1 Concluded d.) e.) f.) Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions DEFINITION: For any positive number x, Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions THEOREM 4: Properties of Natural Logarithms SAME AS log x Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 3: Solve for t. Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions THEOREM 5 ln x exists only for positive numbers x. The domain is (0, ∞). ln x < 0 for 0 < x < 1. ln x = 0 when x = 1. ln x > 0 for x > 1. The function given by f (x) = ln x is always increasing. The range is the entire real line, (–∞, ∞). Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions THEOREM 6 For any positive number x, Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Quick Check 2 Differentiate: a.) , b.) , c.) , Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions THEOREM 7 or The derivative of the natural logarithm of a function is derivative of the function divided by the function. Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 4: Differentiate Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 4 (concluded): Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Quick Check 3 Differentiate: a.) b.) c.) d.) Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 5: In a psychological experiment, students were shown a set of nonsense syllables, such as POK, RIZ, DEQ, and so on, and asked to recall them every second thereafter. The percentage R(x) who retained the syllables after t seconds was found to be given by the logarithmic learning model: a) What percentage of students retained the syllables after 1 sec? b) Find R(2), and explain what it represents. Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Example 5 (concluded): a) b) The result indicates that 2 seconds after students have been shown the syllables, the percentage of them who remember the syllables is shrinking at the rate of 13.5% per second. Copyright © 2014 Pearson Education, Inc.

3.2 Logarithmic Functions Section Summary A logarithmic function is defined by , for and . The common logarithmic function is defined by , for . The natural logarithm function is defined by , where . The derivative of is , for . The slope of a tangent line to the graph of at is found by taking the reciprocal of the input . Copyright © 2014 Pearson Education, Inc.