3.3 Logarithmic Functions

Graph and apply transformations to the graphs of logarithmic functions.

http://nicktoons.nick.com/videos/clip/stimpys-big-day-log-song-1.html

GRAPH AND APPLY TRANSFORMATIONS TO THE GRAPHS OF LOGARITHMIC FUNCTIONS.

An Example…

Let .

A. Find the domain of .B. Sketch the graph of by hand and give coordinates of the - and - intercepts when they exist.

A. B. C. D.

0% 0%0%0%

In your homework, you learned that logarithmic functions are simply the

inverse of exponential functions. This means that the domain of any logarithm

originates from the __________ of the exponential function that is it’s inverse.

A. domainB. rangeC. zerosD. discontinuities

10

A. B. C. D.

0% 0%0%0%

For any logarithmic function,

the inverse function is.

What is the range of the exponential function ?

10

A. B. C. D. E.

0% 0%0%0%0%

So, the domain of

is all such that…

A. None of the above.10

We know that when

the domain is all such that .

Back to our problem…

Solve for to find the domain of .

A. B. C. D. E.

0% 0%0%0%0%

Find the domain of

A. None of the above.

10

An Example…

Let .

A. Find the domain of .B. Sketch the graph of by hand and give coordinates of the - and - intercepts when they exist.

OK, so the instructions say to sketch by hand, but……instead we are just using the MyLabsPlus graphing system.

First you will have to choose a tool from the palette.

Recall that we are looking for the graph of the inverse of an exponential function.

For each plotted point, that we find on the graph of an exponential function, we find on the graph of the logarithmic function.

A. B. C. D.

0% 0%0%0%

Which of the following is the “Logarithm tool”?

A.

B.

C.

D.

10

Actually, if you’re not sure, MyLabsPlus makes it really easy for you!

Just hover the mouse over each tool and it will pop up with a box that says it’s name.

Click on the tool and then click anywhere in the coordinate plane to launch the input boxes…

The default values for a logarithm function are shown in the blue boxes to the left.They correspond to the function

Compare this to the transformation form of the logarithmic function¿𝐶

¿𝑏

¿ 𝐴

¿𝐵¿𝐷

Check this box if .

Check this box if .

A. B. C. D. E.

0% 0%0%0%0%

Write our function

in the form.

A. None of the above.

10

By comparing

to

we can see that

An Example…

Let .

A. Find the domain of .B. Sketch the graph of by hand and give coordinates of the - and - intercepts when they exist.

A. B. C. D. E.

0% 0%0%0%0%

Finding the -intercept is done the same way for every function, including

logarithms. Which of the following is the method that will find the -intercept?

A. Set the function equal to zero and solve for .

B. Find .C. Solve for . Then replace with D. Apply the quadratic formula.E. None of the above.

10

Recall (from your homework) that is defined as the exponent we must raise to in order to get .

A. B. C. D. E. F.

0% 0% 0%0%0%0%

Find the -intercept.

A. None of the above.

10

HINT:

An Example…

Let .

A. Find the domain of .B. Sketch the graph of by hand and give coordinates of the - and - intercepts when they exist.

A. B. C. D. E.

0% 0%0%0%0%

Finding the -intercept is done the same way for every function, including

logarithms. Which of the following is the method that will find the -intercept?

A. Set the function equal to zero and solve for .

B. Find .C. Solve for . Then replace with D. Apply the quadratic formula.E. None of the above.

10

Solve for .

A. B. C. D. E.

0% 0%0%0%0%

This is not as difficult as it might look.Recall from your homework that

A. None of the above.

10

HINT:

Solve for .

A. B. C. D. E. F.

0% 0% 0%0%0%0%

Find the -intercept.

A. None of the above.

10