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15.2 Logarithmic Functions

15.2 Logarithmic Functions. OBJ:  To find the base-b logarithm of a positive number  To find a base-b logarithm equation. Graph y = 2 x - 3 – 1. Domain: Range:  (-1, ) 2. Hor / Vert 3 R, 1  3. Inc/Dec Inc Key Point ( , ) (3, 0)

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15.2 Logarithmic Functions

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  1. 15.2 Logarithmic Functions OBJ:  To find the base-b logarithm of a positive number  To find a base-b logarithm equation

  2. Graph y = 2 x - 3 – 1 • Domain: Range:  (-1, ) 2. Hor/Vert 3 R, 1 3. Inc/Dec Inc • Key Point ( , ) (3, 0) • Equation of asymp. y = -1

  3. Graph y = 2 x - 3 – 1 • Domain: Range:  (-1, ) 2. Hor/Vert 3 R, 1 3. Inc/Dec Inc • Key Point ( , ) (3, 0) • Equation of asymp. y = -1

  4. Graph y = log 2 ( x + 3 ) – 2 • Domain: Range: (-3, )  2. Hor/Vert 3 L 2  3. Inc/Dec Inc • Key Point ( , ) (-2, -2) 5. Equation of asymp. x = -3

  5. Graph y = log 2 ( x + 3 ) – 2 • Domain: Range: (-2, )  2. Hor/Vert 3 L 2  3. Inc/Dec Inc • Key Point ( , ) (-2, -2) 5. Equation of asymp. x = -3

  6. 9 = r – 2/3 32 = r – 2/3 (32)-3/2 = (r – 2/3)-3/2 3 -3 = r 1/27 = r z 5/2 = 243 z 5/2 = 3 5 (z5/2)2/5= (3 5)2/5 z = 32 z = 9 Solve. HW 3 (35-38)

  7. EX:  log 3 9 log 3 9 = x 3 x = 9 3 x = 3 2 x = 2 log 3 81 log 3 81 = x 3 x = 81 3 x = 3 4 x = 4  log 3 1 log 3 1 = x 3 x = 1 3 x = 3 0 x = 0  log 3 1/3 log 3 1/3 = x 3 x = 1/3 3 x = 3 -1 x = - 1 DEF: Base – b logarithmIf y = b x , then log b y = xIf answer = b exponent , then log base answer = exponent

  8. log 3 3 log 3 3 = x 3 x =  3 3 x = 3 ½ x = ½ log 3 1/27 log 3 1/27 = x 3 x = 1/27 3 x = 3-3 x = -3 EX: 1 log 10 100 P 392 log 10 100 = x 10 x = 100 10 x = 102 x = 2 log 21/16 log 21/16 = x 2 x = 1/16 2 x = 2- 4 x = - 4 EX 2  log 34 3 P 392 log 34 3 = x 3 x = 4 3 3 x = 3 ¼ x = 1/4  log 25 8 log 25 8 = x 2x = (23)1/5 x = 3/5

  9. EX:  log 35 81 log 35 81 = x 3x = 5 81 3x = 34/5 x = 4/5 log 1/3 1/9 log 1/3 1/9 = x (1/3)x = 1/9 (1/3)x = (1/3)2 x = 2 log 1/3 9 log 1/3 9 = x (1/3)x = 9 3-x = 32 x = -2 EX:  log b 10,000 = 4 b4 = 10,000 b4 = 104 b = 10  log 3 y = 5 35 = y 243 = y EX:  1/81 = b– 4 3– 4 = b– 4 3 = b 1/8 = b – 3 (2)– 3= b– 3 2 = b Find each logarithm. Solve each equation.

  10. EX: 3 log b 81 = 4 P393b 4 = 81 b 4 = 3 4 b = 3 log 5 625 = x 5 x = 625 5 x = 5 4 x = 4 log 4 y = 3 4 3 = y y = 64 log b1/8 = - 3 b -3 = 1/8 b -3 = 2 -3 b = 2 Solve each logarithmic equation for the variable.HW4 P393 (1-28 all)

  11. EX:  log 5 625 log 5 625 = x 5 x = 625 5 x = 5 4 x = 4  log 7 1 log 7 1 = x 7 x = 1 7 x = 7 0 x = 0  log 7 7 log 7 7 = x 7 x = 7 7 x = 7 1 x = 1 log 5 5 log 5 5 = x 5 x = 5 ½ x = ½ Find each logarithm.

  12. P 394 If y = 2 x what is y – 1 (y inverse)? x = 2 y log2x = log 22 y log2x = y y = log2x DEF:  Inverse of an exponential function y = logbx NOTE: Since y = 3 x and y = log 3 x are inverse functions, they are symmetric with respect to y = x and the graph of y = 3 x could be reflected in y = x to obtain y = log 3 x

  13. EX:  a = log b c; t = b v b a = c; v = log b t 4 = log 2 16 2 4 = 16 9/16 = (3/4)2 2 = log ¾ 9/16  5 = log b 70 b5 = 70  60 = 4 q q = log 4 60 Rewrite the logarithmic equations in exponential form and the reverse procedure.

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