1. 7-3 Logarithmic Functions Warm Up Use mental math to evaluate. 1. 4 –3 3. A power has a base of –2 and exponent of 4. Write and evaluate the power. 2. 7.3.1: Write equivalent forms for exponential and logarithmic functions. 7.3.2: Write, evaluate, and graph logarithmic functions. LEARNING GOALS – LESSON 7.3 A snow day phone tree works as follows; In the first round of calls the principal and two assistant principals call 2 teachers each. Each teacher calls 2 more teachers and so on and so forth. The situation is modeled by the following function: y = _____(_____) x where y is the number of teachers contacted and x is the round of calls. After how many rounds are all 300 teachers reached? _______ = _______ The inverse operation of taking something to the x power is a ___________________. A _________________ is the exponent to which a specified base is raised to obtain a given value. 3 5 = 243

2. 7-3 Logarithmic Functions Write each exponential equation in logarithmic form. Example 1: Converting from Exponential to Logarithmic Form Exponential Equation Logarithmic Form 25 = 5 2 10 4 = 10,000 3 -1 = ⅓ a c = d Exponential Equation Logarithmic Form 9 2 = 81 3 3 = 27 x 0 = 1(x ≠ 0) Check Your Understanding Example 2: Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation log 9 9 = 1 log 2 512 = 9 log 8 2 = ⅓ Log 2 ¼ = –2 log b 1 = 0 Logarithmic Form Exponential Equation log 10 10 = 1 log 12 144 = 2 log ½ 8 = –3 Check Your Understanding

3. 7-3 Logarithmic Functions Log Rule:Exponents: log b b = 1b 1 = b log b 1 = 0b 0 = 1 A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10. For example: Evaluate the following logs. A. log 5 5 = B. log 5 1 = Evaluate by using mental math. Example 3A: Evaluating Logarithms by Using Mental Math A. log 0.01 B. log 5 125 C. log 5 ⅕ E. log 25 0.04 D. log 0.00001

4. 7-3 Logarithmic Functions Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2 x, is a logarithmic function, such as y = log 2 x. y = 2 x DOMAIN: {R} all real numbers RANGE: {y|y > 0} positive numbers y = log 2 x DOMAIN: {x|x > 0} positive numbers RANGE: {R} all real numbers Example 4A: Graphing Logarithmic Functions Use x = –2, –1, 1, 2, and 3 to graph f(x) = ¾ x. Then graph its inverse. Describe the domain and range of the inverse function. x–2–1123 f(x) = ¾ x * Use table feature on calculator to fill in table. x f(x) -1 = log ¾ x RECALL WHAT YOU KNOW ABOUT INVERSES!

5. 7-3 Logarithmic Functions The table lists the hydrogen ion concentrations for a number of food items. Find the pH of each. Example 5: Food Application SubstanceH + conc. (mol/L) Milk 0.00000025 Tomatoes 0.0000316 Lemon juice 0.0063 Milk The hydrogen ion concentration is 0.00000025 moles per liter. FORMULA: pH = –log[H + ] pH = –log(_______________) Substitute the known values in the function. Tomatoes Lemon juice