Brief 3.1 Review Exponential Functions Exponential Functions Additional Example Use transformation of 𝑦 = 2𝑥 to graph the function. y 𝑓 𝑥 = −2𝑥−1 5 Asymptote(s): x Domain: Range: -5 -5 5 The Natural Base 𝒆 • Irrational number (transcendental actually . . . like 𝜋) • 𝑒 is called the “natural base” and 𝑓 𝑥 = 𝑒 𝑥 is called the “natural exponential function.” • 𝑒 ≈ 2.718281827 • Formal Definition of 𝑒: 1 lim 1 + 𝑛→∞ 𝑛 𝑛 Section 3.2 Logarithmic Functions 𝑥 What’s the inverse of 𝑓 𝑥 = 2 ? y 5 x -5 -5 5 Formal Definition Example 1 Write each equation in its equivalent exponential form: a. 2 = log 5 𝑥 b. 3 = log 𝑏 64 c. log 3 7 = 𝑦 Your Turn Write each equation in its equivalent exponential form: a. 3 = log 7 𝑥 b. 2 = log 𝑏 25 c. log 4 26 = 𝑦 Example 2 Write each equation in its equivalent logarithmic form: a. 122 = 𝑥 b. 𝑏 3 = 8 c. 𝑒 𝑦 = 9 Your Turn Write each equation in its equivalent logarithmic form: a. 25 = 𝑥 b. 𝑦 9 = 216 c. 𝑒 𝑥 = 33 Example 3 Evaluate. a. log 2 16 b. 1 log 7 49 Example 3 (cont.) c. log 25 5 5 d. log 2 2 e. 1 log 3 3 Some Basic Properties Example 4 Evaluate: a. log 7 7 b. log 5 1 Remember . . . • The logarithmic function is the inverse of the exponential function. f(f 1 ( x)) f 1 ( f ( x)) x Example 5 Evaluate. a. log 4 45 b. 6log6 9 Graphs of Logarithmic Functions Asymptote(s): Domain: Range: Two Cases 1. 𝑏 > 0 Two Cases (cont.) 2. 0 < 𝑏 < 1 Example 6 Use transformations to graph the function. 𝑓 𝑥 = log 2 𝑥 − 1 − 2 y 5 Asymptote(s): Domain: x Range: -5 -5 5 Example 7 Find the domain. 𝑓 𝑥 = log 4 (𝑥 + 3) Common Logarithms • The logarithmic function with base 10 is called the common logarithmic function. • Typically, log10 𝑥 is just expressed log 𝑥. • On your calculator, hit the button. Example 8 Evaluate without a calculator. a. log 1000 b. 10log 51 c. log .01 Natural Logarithms • The logarithmic function with base 𝑒 is called the natural logarithmic function. • Typically, log 𝑒 𝑥 is just expressed ln 𝑥. • On your calculator, hit the button. Example 9 Evaluate without a calculator. a. ln 𝑒 b. 𝑒 𝑥+1 ln 5𝑥 2 c. log .01 Example 10 Use transformations to graph the function. 𝑓 𝑥 = ln −𝑥 − 2 y 5 Asymptote(s): Domain: x Range: -5 -5 5 Example 11 Find the domain. 𝑓 𝑥 = ln(3 − 𝑥) Example 12 Find the domain. 𝑓 𝑥 = ln(𝑥 − 3)2 True or False? a. log2 8 log2 4 = 8 4 True or False b. log −100 = −2 True or False c. log 𝑏 𝑥 is the exponent to which 𝑏 must be raised to get 𝑥. True or False Questions??? Make sure to be working in MyMathLab!!!