Chapter 3 Exponential and Logarithmic Functions Pre-Calculus Chapter 3 Exponential and Logarithmic Functions

3.2 Logarithmic Functions Objectives: Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions. Recognize, evaluate, and graph natural logarithmic functions. Use logarithmic functions to model and solve real-life problems.

What is a logarithm? “Logarithm” comes from two Greek words “Logos” = ratio “Arithmos” = number So, a logarithm is an exponent.

How Were Logarithms Developed? Imagine a time before calculators…. In the 16th and 17th centuries, scientists made significant gains in the study of astronomy, navigation, and other areas. They wanted an easier (and less error-prone) way of performing calculations, esp. multiplication and division. Two mathematicians, John Napier and Henry Briggs, are credited with developing logarithms to simplify calculations.

How Were Logarithms Used? Recall the property of exponents: ax · ay = a(x + y) Napier used this property to convert complicated, difficult multiplication problems into easier, less error-prone, addition problems. For example, 4,971.26 x 0.2459 = 10m x 10n = 10(m + n) where 3 < m < 4 and –1 < n < 0 Briggs spent a great deal of his life identifying values of y such that 10y = x. The value y is the logarithm. That is, y = log10 x. This data was organized into logarithmic tables.

A Sample Table Log. Exponent Form Number 100 1 log101 = 0 0.08720 100 1 log101 = 0 0.08720 100.08720 1.222 log101.222 = 0.087 0.39076 100.39076 2.459 log102.459 = 0.39076 0.69644 100.69644 4.971 log104.971 = 0.69644 101 10 log1010 = 1

Back to the Example Original statement: 4,971.26 x 0.2459 Rewrite in scientific notation: 4.97126 x 103 x 2.459 x 10-1 Use values from table: 100.69644 x 103 x 100.39076 x 10-1 Simplify: 103.08720 103 x 100.08720 1, 000 x 1.222 = 1,222 …Fortunately, we have calculators!

Definition of Logarithmic Function The logarithmic function with base a is given by f (x) = loga x where x > 0, a > 0, and a ≠ 1. That is, y = loga x if and only if x = a y

Example 1 Rewrite each as an exponential expression and solve. f (x) = log2 32 f (x) = log3 1 f (x) = log4 2 f (x) = log10 1/100

Example 2 Use your calculator to solve f (x) = log10 x at each value of x. x = 10 x = 2.5 x = –2 x = ¼

Properties of Logarithms loga 1 = 0 because a0 = 1. loga a = 1 because a1 = a. loga ax = x and a loga x = x Inverse Properties If loga x = loga y, then x = y. One-to-One Property

Example 3 Solve for x: log2 x = log2 3 Solve for x: log4 4 = x Simplify: log5 5x Simplify: 7 log7 14

Graph of Logarithmic Function   To sketch the graph of y = loga x, first graph y = ax, then graph its inverse. Example: Graph f (x) = log2 x. Construct a table of values for g(x) = 2x and plot the points. The function f (x) = log2 x is the inverse of g(x) = 2x. Note: Inverse functions are reflections in the line y = x.

Logarithmic Function The logarithmic function is the inverse of the exponential function. Many real-life phenomena with a slow rate of growth can be modeled by logarithmic functions.

Characteristics of the Logarithmic Function

Transformations of f (x) = loga x The graph of g(x) = loga (x ± h) is a _________ shift of f. The graph of h(x) = loga (x) ± k is a _________ shift of f. The graph of j(x) = – loga (x) is a reflection of f ______ . The graph of k(x) = loga (– x) is a reflection of f ______ .

Example 4   Each of the following functions is a transformation of the graph of f (x) = log10 x. Describe the transformation and graph. g(x) = log10 (x – 1) h (x) = 2 + log10 x

Natural Logarithmic Function For x > 0, y = ln x if and only if x = e y The function given by f (x) = loge x = ln x is called the natural logarithmic function.

Example 5 Use your calculator to solve f (x) = ln x at each value of x. x = 2 x = 0.3 x = –1

Properties of Natural Logs ln 1 = 0 because e0 = 1. ln e = 1 because e1 = e. ln ex = x and e ln x = x. Inverse Properties If ln x = ln y, then x = y. One-to-One Property

Example 6 Use properties of natural logs to rewrite each expresssion.

Graph of Natural Log Function The graph of y = ln x is the reflection of the graph y = ex in the line y = x.

Domain of Log Functions Find the domain of each function. f (x) = ln (x – 2) g(x) = ln (2 – x) h(x) = ln x2

Transformations of f (x) = ln x The graph of g(x) = ln (x ± h) is a ___________ shift of f. The graph of h(x) = ln (x) ± k is a ___________ shift of f. The graph of j(x) = – ln (x) is a reflection of f ________. The graph of k(x) = ln (– x) is a reflection of f ________.

Example 7 Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model f (t) = 75 – 6 ln (t + 1), 0 ≤ t ≤ 12 where t is the time in months. What was the average score on the original (t = 0) exam? What was the average score at the end of t = 2 months? What was the average score at the end of t = 6 months?

Homework 3.2 Worksheet 3.2