The logarithm as a function
revision
Match the function to graph and then describe the transformation.
y = f(x) + 3
y = 2f(x)
y = f(-x)
y = f(x-2)
y = -f(x)
y = f( ½ x)
y = ½ f(x)
y = f(2x)
recap
Give possible values
log 2 8 = 3
The number of cells, n at time t = 0 is n = 1. The number of cells, n at time t = 1 is n = 2.
Write a function for n in terms of t.
Describe how the number of cells are growing.
Sketch a graph of n against t.
Imagine plotting a graph of y=2x, with 1cm to one unit on each axis.
How far along the x-axis could you go before the graph reached the top of a sheet of paper?
If you extended the graph so the positive x-axis filled the whole width of a sheet of paper, how tall would the paper have to be?
How far along the x-axis would you have to go so that the graph was tall enough to reach
Try to estimate the answers before calculating them and mark them at the appropriate points along a sketch of the x-axis.
Work out where they should be and then add some other results such as the distances to the sun and other stars. What do you notice?
Data you may find useful is on the following slide.
enter y = ax
click ‘add slider’
Describe the shape of the graph. Sketch the curves.
What do the curves have in common?
What happens as a increases?
Is the curve defined for all values of a?
If not, why not?
What happen between a = 0 and a = 1?
To a draw a series of these curves:
Investigate graphs of the form y = kax
enter y = k2x
click ‘add slider’
Describe the shape of the graph. Sketch the curves.
What do the curves have in common?
What happens as k increases?
What happens as k decreases?
How is y = 2x transformed by y=k2x?
To a draw a series of these curves:
Investigate graphs of the form y = acx
enter y = 2cx
click ‘add slider’
Describe the shape of the graph. Sketch the curves.
What do the curves have in common?
What happens as c increases?
What happens as c decreases? When c is negative?
How is y = 2x transformed by y=2cx?
To a draw a series of these curves:
Compare y=2x and y = (½)x
Compare y=ax and y = (1/a)x
Explain.
1
2
Watch the video
The function y = loga x
Complete
y = loga x ⇔ x = ay
Complete
y = log2 x ⇔ x = 2y
y | x = 2y |
0 | |
1 | |
2 | |
3 | |
4 | |
Sketch x = 2y
Plot the graphs of y = 2x and x = 2y.
What do you notice?
Investigate y = ax and x = ay
Complete
x | 2x | log 2 2x |
0 | | |
1 | | |
2 | | |
3 | | |
4 | | |
x | 3x | log 3 3x |
0 | | |
1 | | |
2 | | |
3 | | |
4 | | |
x | 10x | log 10 10x |
0 | | |
1 | | |
2 | | |
3 | | |
4 | | |
What does this suggest?
Investigate the functions y = ax and y = loga x
Driving the point home
Open desmos and follow the steps in order
1. Plot f(x) = 2x
2. Plot a point, p on the curve with coordinates (p, f(p)).
Add a slider - move it.
3. Delete the slider.
Reflect the point, p in the line y = x by plotting (f(p), p).
Add a slider - move it.
4. Delete the slider and plot a series of points. What do you notice?
5. Plot y = log2 x.
What do you conclude?
Watch the video
Betweener
The curve shown�is y = 2x.
m is an integer.
What is the �value of m?��
Log region
Each straight�line shown is �either horizontal �or vertical.��What is the area�of the rectangle �shown?
Big - L
The point A has�coordinates (a, a) where�a = log2(3).
What is the area of the �shaded L shape �in terms of a?��
extension
Carefully copy into desmos :
What do the graphs show?
Vary p.
Find a link between the gradient of the tangent to (p, f(p)) on f(x) = ax and the gradient of the tangent to (f(p), p) on g(x) = loga x.