Exploring Logarithm graphs using GeoGebra
1. In the box at the bottom, in order to enter y=log10(x), type “y=log(10,x)”.
a) For what x value is log10(x)=0?
b) Write an equation of the form 10a = b to explain why this is the correct x value.
c) For what x value is log10(x)=1?
d) Write an equation of the form 10a = b to explain why this is the correct x value.
2. Use the left arrow key to move your graph to the left. If you kept going, would the y
value keep getting larger or stop below some number? Record your guess here.
a) What y value would you get if you plugged in x=100?
b) What y value would you get if you plugged in x=1000?
c) As the x values get larger and larger, do the answers get larger and larger, or
do they stop at some point?
d) Do you want to change your guess?
3. Now type in y=log(2,x) in the box at the bottom. This gives the equation y=log2(x).
a) How does this graph compare to the other graph?
b) Explain what you notice about the two graphs.
c) Where do the two graphs cross?
4. Click on the first graph (the graph of y=log(10,x)) and delete it.
Next, click on the New Point button (on the left side), and click on the other graph to add
a point you can drag around. You can drag that point to answer these questions.
a) For what x value is log2(x)=-1?
b) Write an equation of the form 2a = b to explain why this is the correct value.
c) For what x value is log2(x)=-2?
d) Write an equation of the form 2a = b to explain why this is the correct value.
5. Also plot y=log2(2x) and add a point to that graph.
a) By replacing x with 2x, what happens to the graph?
b) It also appears that y=log2(2x) is simply y=log2(x), shifted up. About how
much is it shifted up?
c) In order to be sure, how can you use log rules to show this for sure?
Bonus problems:
6) Delete the plots, and instead graph y=log(2,x), y=log(2,x2), y=log(2,x3). Put a point on
each of these curves at the same x-value.
a) What do you notice about the y-values of these points?
b) Explain why this is by using log rules.
7) Suppose an equation of the form y=a log2(x)+b goes through (8,0) and (4,-2). What is
each question mark in this equation? Check your equation goes through these two points.