MAT 125 – Applied Calculus Logarithmic Functions

Today’s Class  We will be learning the following concepts today:  Exponential Functions & Their Graphs  Logarithmic Functions & Their Graphs  Properties Relating Exponential and Logarithmic Functions  Compound Interest  Effective Rate of Interest  Continuous Compounding of Interest Dr. Erickson 5.2 Logarithmic Functions 2

Introduction  You are already familiar with exponential equations of the form b y = x (b > 0, b ≠ 1) where the variable x is expressed in terms of a real number b and a variable y.  If we solve this for y, then the number y is called the logarithm of x to the base b and is denoted by log b x. Dr. Erickson 5.2 Logarithmic Functions 3

4 Definition Dr. Erickson

The two most widely used symbols of logarithms are the system of common logarithms, which uses the number 10 as its base, and the system of natural logarithms, which uses the irrational number e ≈ … as its base. 5.2 Logarithmic Functions 5 Logarithms Dr. Erickson

Note  The system of natural logarithms is used in theoretical work. Using natural logarithms rather than logarithms to other bases often leads to simpler expressions. Dr. Erickson 5.2 Logarithmic Functions 6

Laws of Logarithms Dr. Erickson 5.2 Logarithmic Functions 7

Properties Dr. Erickson 5.2 Logarithmic Functions 8

Example  Express each equation in logarithmic form. Dr. Erickson 5.2 Logarithmic Functions 9

Example  Write the expression as the logarithm of a single quantity. Dr. Erickson 5.2 Logarithmic Functions 10

Example  Use the laws of logarithms to expand and simplify the expression. Dr. Erickson 5.2 Logarithmic Functions 11

Example  Use logarithms to solve the equation for t. Dr. Erickson 5.2 Logarithmic Functions 12

Example Dr. Erickson 5.2 Logarithmic Functions 13

Let’s continue with 5.3 Dr. Erickson 5.2 Logarithmic Functions 14