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Definition of a Logarithmic Function

Exponent. Exponent. Base. Base. Logarithmic: y = log b x Exponential: b y = x . Form Form. Date: 3.3: Logarithmic Functions and Their Graphs (3.3). Definition of a Logarithmic Function. For x > 0 and b > 0, b≠ 1, y = log b x is equivalent to b y = x

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Definition of a Logarithmic Function

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  1. Exponent Exponent Base Base Logarithmic: y = logbx Exponential: by = x. Form Form Date:3.3: Logarithmic Functions and Their Graphs (3.3) Definition of a Logarithmic Function For x > 0 and b > 0, b≠ 1, y = logb x is equivalent to by = x The function f (x) = logb x is the logarithmic function with base b.

  2. Write each equation in its equivalent exponential form. a. 3 = log7x b. 2 = logb 25 c. log4 26 = y Text Example or y = log4 26 73 = x b2 = 25 4y = 26 y = logb x means by = x Write each equation in its equivalent logarithmic form. a. 122 = x b. b3 = 8 c. ey = 9 2 = log12x 3 = logb 8 y = loge 9

  3. Logarithmic Expression Question Needed for Evaluation Logarithmic Expression Evaluated a. log2 16 b. log3 9 c. log25 5 Text Example Evaluate a. log2 16 b. log3 9 c. log25 5 = 1/2 = 4 = 2 Solution 2 to what power is 16? log2 16 = 4 because 24 = 16. log3 9 = 2 because 32 = 9. 3 to what power is 9? log25 5 = 1/2 because 251/2 = 5. 25 to what power is 5?

  4. Basic Logarithmic Properties Involving One logbb = 1 because 1 is the exponent to which b must be raised to obtain b (b1 = b) logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1 (b0 = 1)

  5. For x > 0 and b  1, logb by= y The logarithm with base b of b raised to a power equals that power. b logb x = x b raised to the logarithm with base b of a number equals that number. Inverse Properties of Logarithms

  6. y = x 6 5 f (x) = 2x 4 3 2 g (x) = log2x -2 -1 2 3 4 5 6 -1 -2 Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx • The x-intercept is 1. There is no y-intercept. • The y-axis is a vertical asymptote. • If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing. • The graph is smooth and continuous. It has no sharp corners or edges.

  7. Reverse coordinates. x -2 -1 0 1 2 3 x 1/4 1/2 1 2 4 8 6 5 4 f (x) = 2x 1/4 1/2 1 2 4 8 g(x) = log2 x -2 -1 0 1 2 3 y = x 3 f (x) = 2x 2 -2 -1 2 3 4 5 6 -1 g (x) = log2x -2 Text Example Graph f (x) = 2x and g(x) = log2x in the same rectangular coordinate system. Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2x. Graph f (x) = 2x and g(x) = log2x in the same rectangular coordinate system. The graph of the inverse can also be drawn by reflecting the graph of f(x) = 2x over the line y = x.

  8. Complete Student Checkpoint Write each equation in its equivalent logarithmic form. a. 25 = x b. b3 = 27 c. ey = 33 5 = log2x 3 = logb 27 y = loge 33 Evaluate a. log10 100 b. log3 3 c. log36 6 = 1/2 = 2 = 1 36? = 6 10? = 100 3? = 3

  9. h(x) h(x) x y 1/9 -2 1/3 -1 1 0 3 1 9 2 Complete Student Checkpoint and on the same graph Graph For h(x) just interchange the x and y values of f(x) f(x) f(x) x y -2 1/9 -1 1/3 0 1 1 3 2 9

  10. Finding Domains What is the Domain for the following functions? (remember you can’t take the log of a negative number) f(x) = log2x x > 0 x > -3 g(x) = log4(x+3) x > 1 g(x) = log2(x-1) x > 2 g(x) = log7(x-2) x < 0 h(x) = log2(-x) x < 3 g(x) = ln(3-x) x > 0 j(x) = - log2x g(x) = ln(x-2)2 x > 0 p(x) = log2x + 1 all real numbers x > 0 r(x) = 2log2x Complete Student Checkpoint Find the domain of x - 5 >0 x > 5

  11. Transformation Equation Description Horizontal translation • Shifts the graph of f (x) = logbx to the • left c units if c > 0, asymptote x = -c • Shifts the graph of f (x) = logbx to the • right c units if c < 0,asymptote x = c g(x) = logb(x+c) Vertical stretching or shrinking g(x) = clogbx • Multiply y-coordintates of f (x) = logbx by c, • Stretches the graph of f (x) = logbx if c > 1. • Shrinks the graph of f (x) = logbx if 0 < c < 1. Reflecting • Reflects the graph of f (x) = logbx about the x-axis. • Reflects the graph of f (x) = logbx about the y-axis. g(x) = - logbx g(x) = logb(-x) Vertical translation • Shifts the graph of f (x) = logbx upward c units if c > 0. • Shifts the graph of f (x) = logbx downward c units if c < 0. g(x) = logbx+c Transformations Involving Logarithmic Functions

  12. x y 1/2 1 2 4 Text Example Graph f (x) = log2x and g(x) = 1 + -2log2(x - 1) in the same rectangular coordinate system. Right 1 What are the transformations for g(x)? Stretch x2 Reflect over x-axis f(x) Up 1 f(x) -1 0 1 2 g(x)

  13. General PropertiesCommon Logarithms 1. logb 1 = 0 1. log 1 = 0 2. logbb = 1 2. log 10 = 1 3. logbbx = x 3. log 10x = x 4. b logb x = x 4. 10 log x = x Properties of Common Logarithms(base is 10, log10 ) The Log on your calculator is base 10

  14. log b b = Examples of Logarithmic Properties 1 3 log 3 6 = 6 log b 1 = 0 log 5 5 3 = 3 log 4 4 = 1 2 log 2 7 = 7 log 8 1 = 0

  15. General PropertiesNatural Logarithms 1. logb 1 = 0 1. ln 1 = 0 2. logbb = 1 2. ln e = 1 3. logbbx = x 3. ln ex = x 4. b logb x = x 4. eln x = x Properties of Natural Logarithmsln = loge

  16. ln e = Examples of Natural Logarithmic Properties 1 e ln 6 = 6 e log e 6 loge e ln e 3 = 3 ln1 = 0 log e 1 log e e 3 Complete Student Checkpoint Use inverse properties to solve:

  17. Logarithmic Functions and Their Graphs

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