Logarithmic Functions. Objective To graph logarithmic functions To graph logarithmic functions To evaluate logatrithms To evaluate logatrithms.

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Presentation transcript:

Logarithmic Functions

Objective To graph logarithmic functions To graph logarithmic functions To evaluate logatrithms To evaluate logatrithms

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

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

**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!

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

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

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

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

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

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

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

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

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 is the vertical asymptote. is the x-intercept Since this is not possible, there is No y-intercept.

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

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.

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

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

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.