Comparing Logarithm and Exponential Functions Graphs

Change the position of the slider in order to change the base [b][i]a[/i][/b] which is used as the base of the exponential and the logarithmic functions.

Uncheck the checkbox so the logarithmic graph is hidden and only the exponential graph is showing.

Task 1

Recall the properties of an exponential graph. Select all that apply for [math]y=a^x[/math] where [math]a>1[/math]

The range is y > 0 As [math]x\rightarrow-\infty[/math], [math]f\left(x\right)\rightarrow0[/math] As [math]x\rightarrow-\infty[/math], [math]f\left(x\right)\rightarrow\infty[/math] The vertical asymptote is x=0 The domain is all real numbers The horizontal asymptote is y=0 As [math]x\rightarrow\infty[/math], [math]f\left(x\right)\rightarrow0[/math] The y-intercept is (0,1) The x-intercept is (1,0) The range is all real numbers The domain is x > 0 As [math]x\rightarrow\infty[/math], [math]f\left(x\right)\rightarrow\infty[/math]

Now uncheck the Exponential graph and select the Logarithmic graph.

Task 2

Notice the properties of the logarithmic graph. Select all that apply for [math]y=log_ax[/math] where [math]a>1[/math]

The range is all real numbers As [math]x\rightarrow\infty[/math], [math]f\left(x\right)\rightarrow\infty[/math] As [math]x\rightarrow0[/math], [math]f\left(x\right)\rightarrow-\infty[/math] As [math]x\rightarrow\infty[/math], [math]f\left(x\right)\rightarrow0[/math] The range is y > 0 The vertical asymptote is x=0 The y-intercept is (0,1) The domain is x > 0 The horizontal asymptote is y=0 The domain is all real numbers As [math]x\rightarrow0[/math], [math]f\left(x\right)\rightarrow\infty[/math] The x-intercept is (1,0)

Make both the Exponential and Logarithmic graphs visible. Locate point D on the exponential function and point B on the logarithmic function.

Task 3

What is the relationship between points D and B?

Task 4

How do the properties of the two function types relate? Select all that apply.

Nothing about them is the same. They are both smooth and continuous They both start at x=0 Adding a constant to the end of either one will result in a vertical shift down. They both go to infinity as [math]x\rightarrow\infty[/math]

Task 5

How do the properties of the two function types differ? Select all that apply.

One has an x-intercept while the other has a y-intercept. They do not differ, they have the same properties. One is completely below the x-axis while the other is completely above the x-axis. One has a horizontal asymptote while the other has a vertical asymptote.

Task 6

What do you notice about the intersection between the exponential and logarithmic graphs when they have the same base?

They sometimes intersect. They always intersect. Their point of intersection lies on the x-axis. Their point of intersection lies on the line line y=x. They never intersect.

Task 7

What is the slope and y-intercept of the line y=x? Select multiple answers.

The y-intercept is (0,1) The slope is 0. The y-intercept is (0,0) The slope is undefined. It has no y-intercept. The slope is 1.

Task 8

If you shift an exponential function horizontally or vertically, how would this transformation apply to its corresponding logarithmic equation? Talk about it with your partner and make a prediction.

Test your conjecture from Task 8 here by adjusting the sliders c and d

Task 9

Did your conjecture hold true? Explain.

Task 10

Fill in the blanks below:When the bases are the same, a [b][u]horizontal [/u][/b]translation of an exponential function corresponds to a ___________ translation of its corresponding logarithmic function.[br]When the bases are the same, a [b][u]vertical [/u][/b]translation of an exponential function corresponds to a ___________ translation of its corresponding logarithmic function.