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Logarithmic Functions

Logarithmic Functions. Objective. To graph logarithmic functions To evaluate logatrithms. The inverse function of an exponential function is the logarithmic function. For all positive real numbers x and b, b>0 and b 1, y=log b x if and only if x=b y .

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Logarithmic Functions

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  1. Logarithmic Functions

  2. Objective • To graph logarithmic functions • To evaluate logatrithms

  3. The inverse function of an exponential function is the logarithmic function. • For all positive real numbers x and b, b>0 and b1, y=logbx if and only if x=by. • The domain of the logarithmic function is

  4. The range of the function is • Since the log function is the inverse of the exponential function, their graphs are symmetric with respect to the line y=x.

  5. **Remember that since the log function is the inverse of the exponential function, we can simply swap the x and y values of our important points!

  6. Logarithmic Functions where b>1 are increasing, one-to-one functions. • Logarithmic Functions where 0<b<1 are decreasing, one-to-one functions. • The parent form of the graph has an x-intercept at (1,0) and passes through (b,1) and

  7. There is a vertical asymptote at x=0. • The value of b determines the flatness of the curve. • The function is neither even nor odd. There is no symmetry. • There is no local extrema.

  8. More Characteristics of • The domain is • The range is • End Behavior: • As • As • The x-intercept is • The vertical asymptote is

  9. There is no y-intercept. • There are no horizontal asymptotes. • This is a continuous, increasing function. • It is concave down.

  10. Graph: Domain: Range: x-intercept: Important Points: Vertical Asymptote: Inc/dec? increasing Concavity? down

  11. Graph: *Reflects @ x-axis. Domain: Range: x-intercept: Vertical Asymptote: Inc/dec? decreasing Concavity? up Important Points:

  12. Transformations Horizontal shift right 1. Reflect @ x-axis. * Reflect @ x-axis. Vertical shift up 2. Vertical stretch of 2. Vertical shift up 1. Vertical shift down 3. Domain: Range: x-intercept: Domain: Range: x-intercept: Domain: Range: x-intercept: Vertical Asymptote: Vertical Asymptote: Vertical Asymptote: Inc/dec? decreasing Inc/dec? decreasing Inc/dec? increasing Concavity? up Concavity? up Concavity? down

  13. More Transformations Horizontal shrink ½ . Horizontal shrink ½ . Horizontal shift right ½ .. Domain: Range: x-intercept: Domain: Range: x-intercept: Vertical Asymptote: Vertical Asymptote: Inc/dec? increasing Inc/dec? increasing Concavity? down Concavity? down

  14. The asymptote of a logarithmic function of this form is the line To find an x-intercept in this form, let y=o in the equation To find a y-intercept in this form, let x=o in the equation Since this is not possible, there is No y-intercept. is the vertical asymptote. is the x-intercept

  15. Check it out!  is the vertical asymptote. Horizontal shrink ½ . Horizontal shift right 1.  is the x-intercept. Since this is not possible, there is No y-intercept.  Domain: Range: x-intercept: Vertical Asymptote: Inc/dec? increasing Concavity? down

  16. Common Log & Natural Log • A logarithmic function with base 10 is called • a Common Log. • Denoted: • A logarithmic function with base e is called • a Natural Log. • Denoted: *Note there is no base written.

  17. Transformations Common Log Let to find the V.A. Horizontal shrink 1/3. Horizontal shift left 10/3. is the V.A. Let to find the x-intercept. is the x-intercept. Let to find the y-intercept. Domain: Range: Inc/Dec: Concavity: increasing down is the y-intercept.

  18. Transformations Natural Log Let to find the V.A. Reflect @ x-axis. Vertical stretch of 2. Horizontal shrink of ¼ . Horizontal shift left 2. Vertical shift up 2. is the V.A. Let to find the x-intercept. is the x-intercept. Let to find the y-intercept. Domain: Range: x-intercept: Inc/dec? decreasing is the y-intercept. Concavity? up

  19. Change of Base Formula Use this formula for entering logs with bases other than 10 or e in your graphing calculator. So, if you wanted to graph , you would enter in your calculator. Either the natural or common log may be used in the change of base formula. So, you could also enter in Your calculator.

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