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Properties of Logarithms

Properties of Logarithms. 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 3. log a a x = x and a log a x = x inverse property. 4. If log a x = log a y , then x = y . one-to-one property. Examples : Solve for x : log 6 6 = x.

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Properties of Logarithms

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  1. Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. loga ax = x and alogax = x inverse property 4. If logax = logay, then x = y. one-to-one property

  2. Examples: Solve for x: log6 6 = x Simplify: log3 35 Simplify: 7log79 log6 6 = 1 property 2→x = 1 log3 35 = 5 property 3 7log79 = 9 property 3

  3. 4. vertical asymptote y-axis vertical asymptote 2. range 1. domain domain y Graph of f(x) = logax (a> 1) x-intercept (1, 0) x range The graphs of logarithmic functions are similar for different values of a. f(x) = logax (a> 1) Graphs of Logarithmic Functions y = x y = ax y = loga x 3. x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = ax in y = x 3

  4. y-axis vertical asymptote domain y = log4 x y x = 4 y y = x log4 x = y4y= x x-intercept (1, 0) x range The graphs of logarithmic functions are similar for different values of a. f(x) = log4x Graphs of Logarithmic Functions 4

  5. domain y = log4 x y y = x x-intercept (2,0) y = log4 x-1 x range Shifting Graph of Logarithmic Functionf(x) = log4 (x-1) Graphs of Logarithmic Functions 5

  6. domain y = log4 x y y = x+3 (1,2) y = 2 + log4 x x range Shifting Graph of Logarithmic Functionf(x) = 2+log4x Graphs of Logarithmic Functions 2 1 6

  7. Properties of Logarithm

  8. Examples Expand: 1. log3(2x) = log3(2) + log3(x) 2. log4( 16/x ) = log4(16) – log4(x) 3. log5(x3) = 3log5(x) = log2(8) + log2(x4) – log2(5) 4. log2(8x4) – log2(5)

  9. Examples Condense: 1. log2(x) + log2(y) = log2(xy) 2. log3(4) – log3(5) = log3(4/5) log2(x3y) log2((x + 3)4) 3. 3log2(x) – 4log2(x + 3) + log2(y)

  10. Natural Logarithm and e is used to denote

  11. then

  12. = -3 = 3 = -1 = 4 = n =0 =1 =2 = = Without using a calculator find the value of: Using the natural log - ln Evaluate the following ln’s:

  13. The laws of natural logarithms

  14. Find x, if Find x, if Remember the base of a natural log is e. Take a natural log of both sides. Rearrange in index form. Use the power rule. Find x in each of the following:

  15. (0, 1) y The graph of f(x) = ax, a > 1 is INCREASING over its domain Domain: (–∞, ∞) Horizontal Asymptote y = 0 Range: (0, ∞) 4 x 4 The graph of exponential funtion The graph off(x) = ax, a > 1

  16. f(x) = 2x Example 3 Example: Sketch the graph off(x) = 2-x. State the domain and range. y 4 Domain: (–∞, ∞) Range: (0, ∞) x –2 2

  17. (0, 4) f(x) = 2x f(x) = 2x=2 y =0 Range: (0, ∞) Transformation of exponential graphs Example: Sketch the graph off(x) = 2 x + 2 State the domain and range. f(x) = 2 x + 2 y f(x) = 2x ( 22 ) 4 2 f(x) = 2x (4) x Domain: (–∞, ∞) Range: (0, ∞)

  18. f(x) = 2x f(x) = -2x Example 3 Example: Sketch the graph off(x) = -2x. State the domain and range. y 4 x –2 2 Domain: (–∞, ∞) Range: (0, -∞)

  19. Graph of f(x) = ex y 6 4 2 x –2 2 The irrational number e, where e≈ 2.718281828… is used in applications involving growth and decay. Graph of Natural Exponential Function f(x) = ex

  20. y-axis vertical asymptote domain y = log2 x y x = 4 y y = x log4 x = y4y= x x-intercept (1, 0) x range The graphs of logarithmic functions are similar for different values of a. f(x) = log4x Graphs of Logarithmic Functions 20

  21. domain y = log4 x y y = x x-intercept (2,0) y = log4 x-1 x range Shifting Graph of Logarithmic Functionf(x) = log4 (x-1) Graphs of Logarithmic Functions 21

  22. domain y = log4 x y y = x+3 (1,2) y = 2 + log4 x x range Shifting Graph of Logarithmic Functionf(x) = 2+log4x Graphs of Logarithmic Functions 2 1 22

  23. then

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