Table of Contents

Logarithmic Function - Graphing

Example

Sketch the graph of the logarithmic function ...

xlog)x(f 3

• First, write the function in its exponential form.

xlogy 3x3y

• Find a few ordered pairs (plug in y, find x) ...

(1/9, - 2)(1/3, - 1)( 1, 0)( 3, 1)( 9, 2)

and plot them on a graph.-3

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Slide 2

Logarithmic Function - Graphing

• Now join the points with a smooth curve ...• Note that the growth is rapid at first ...and then the rate of growth slows.• Note also that the graph goes through the point (1,0).

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Table of Contents

Logarithmic Function - Graphing

Slide 3

• There is a relationship between the graphs of exponential and logarithmic functions.

Exponential Logarithmicx4y xlogy 4

(0,1) (1,0)(1,4) (4,1)

(2,16) (16,2)

• Note that in the corresponding ordered pairs, the order of x and y is reversed, which suggests inverse functions. In fact, exponential and logarithmic functions to the same base are inverse functions.

Table of Contents Slide 4

Logarithmic Function - Graphing

Exponentialx4y

Logarithmicxlogy 4

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-5 -4 -3 -2 -1 1 2 3 4 5

• To see the inverse relationship graphically, graph each ...

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Table of Contents Slide 5

Logarithmic Function - Graphing

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... and then graph both functions on the same coordinate system.

x4y

xlogy 4

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• Note the symmetry about the line y = x ...

Table of Contents Slide 6

Logarithmic Function - Graphing

• Note the following characteristics of the exponential and the logarithmic functions, which are inverse functions.

Exp Log

• Domain: (- , ) (0, )

• Range: (0, ) (- , )

• Intercept: (0,1) (1,0)

• Asymptote: y = 0Horizontal

x = 0Vertical

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Logarithmic Function - Graphing