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Section 11-4 Logarithmic Functions Objective: Students will be able to

Section 11-4 Logarithmic Functions Objective: Students will be able to Evaluate expressions involving logarithms Solve equations involving logarithms Graph logarithmic functions and inequalities. Example 1: Write each equation in exponential form.

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Section 11-4 Logarithmic Functions Objective: Students will be able to

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  1. Section 11-4 Logarithmic Functions Objective: Students will be able to Evaluate expressions involving logarithms Solve equations involving logarithms Graph logarithmic functions and inequalities

  2. Example 1: Write each equation in exponential form. Since the graphs of exponential functions pass the horizontal line test, we know their inverses are also functions. Remember to find the inverse of a graph, we can switch the x & y variables around and then graph them. The graphs are reflections of each other across the line y = x. y = Inverse x y 0 1 2 -1 x y 0 1 -1 The inverse of an exponential is called a logarithm, and is written as a. log328 = b. = 8 = 3

  3. Example 2:Write each equation in logarithmic form. = = 1296 = Example 3:Evaluate each expression. a. b. log445 = x = x = 5 = x = -3 = x c. log48 d. =

  4. Properties of Logs = , then 3x - 4 = 5x + 2 To Solve Log Equations: Type 1: Logs on both sides Type 2: Logs on one side 1. 1. 2. 2. 3. 3. 4. 4. Simplify to get only one log in the problem. Simplify on both sides to get one log on each side. (Use Prop. Of Logs.) Use Prop. Of Equality and drop the logs. Rewrite problem as an exponent. Solve for the variable. Solve for the variable. Check your answers. Check your answers.

  5. Example 4:Solve each equation. b. = 5x – 3 = 10x + 2 -3 = 5x + 2 -5 = 5x -1 = x P = 81 c. log8(x + 1) + log8 (x + 3) = log8 24 d. log33x = -1 = 3x = 3x + 4x – 21 = 0 = x (x + 7)(x – 3) = 0 x = -7, 3 3

  6. e. log104 + 2 log10 x = 2 f. log3(x + 3) – log3 (2x – 1) = log3 2 = = 2 To Graph Log Functions…. 1) 2) 3) 4) = x Rewrite the logarithm as an exponential Make an xy-table of values for the parent function. Pick numbers for y and solve for x. Shift the parent function based on a, b, h, and k. State the V.A., Domain and Range.

  7. Example 5: Graph y = log4 (x + 2). h = -2 D: x > -2 x y 0 1 -1 x y -1 0 1 - = x R: R V.A. : x = -2 Example 6: Graph y = log3x - 4. k = -4 x y 0 1 -1 x y -4 -3 -5 = x D : x > 0 R : R V.A. : x = 0

  8. Example 7: Graph ln - 3 a = h = -1 k = -3 = x D: x > -1 R: R V.A.: x = -1

  9. Example 8: Graph the following logarithmic inequality.c inequality Parent Graph: = x -2 -1 0 1 9 2 - -2 - -1 0 0 2 1 8 2 Horizontal translation 1 lt. Test (1, 2)

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