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LOGARITHMIC FUNCTIONS

xample 5:

Find the following for .

Remember that can be written as , where the base is the number e.

Domain:

The domain of the logarithmic function consists of all numbers that DO NOT make the y-value imaginary (log of a negative number) or undefined (log of 0).

We do this by finding the numbers that make the argument greater than 0. That is,

x > 0

Therefore, in Interval Notation, the domain is (0, ).

Range:

The function is the inverse of an exponential function. Earlier in the coursewe learned that the domain of a function is the same as the range of its inverse function,while the range of a function is the same as the domain of its inverse function.

We know that all exponential functions have a domain of . Thus, the range of

their inverse functions must also be .

Therefore, we conclude that the range of is .

Intercepts:

The x-intercept (y = 0) expressed as an ordered pair:

We must now solve a logarithmic equation. Changing to exponential form, we find

and

The coordinates of the x-intercept are (1, 0).

The y-intercept (x = 0) expressed as an ordered pair:

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But any logarithm of 0 is undefined. Therefore, we can conclude that this function has NOy-intercepts.

Equation of the Vertical Asymptote:

Since the function is of the the form with b = e, we can conclude that theequation of the vertical asymptoteis

x = 0, which is the y-axis.

Coordinates of several points. Round the y-coordinates to 2 decimal places.

Please note that the points were not arbitrarily selected.

Given a domain greater than 0, we naturally will select positive x-coordinates.In the case of the logarithmic function, we always want to select several x-values very cto the vertical asymptote. As you can see below, the x-values 0.25, 0.5, and 0.75 were

selected in addition to values further away from the vertical asymptote .

(0.25, 1.39), (0.5, 0.69), (0.75, 0.29), (1, 0), (2, 0.69), (4, 1.39), (6, 1.79)

Most logarithmic functions and their transformations are best graphed with a graphing calculatorbecause the y-values get extremely large/small very quickly and are difficult to show in a hand-drawnCartesian Coordinate System.

However, you should be able to say without graphing the function, that the graph has the followingshape, where the y-axis is the vertical asymptote.