6.2: Transformations of the Logarithmic Function

Learning Objectives

For the logarithmic function f(x)=logbx,

  • Perform vertical and horizontal shifts
  • Perform vertical compressions and stretches
  • Perform reflections across the x-axis
  • Perform reflections across the y-axis
  • Determine the transformations of the logarithmic function f(x)=alogb(xh)+k
  • Determine the equation of a function given the transformations
  • Determine what happens to the vertical asymptote as transformations are made

Vertical Shifts

If we shift the graph of the logarithmic function f(x)=log2x up 5 units, all of the points on the graph increase their y-coordinates by 5, but their x-coordinates remain the same. Therefore, the equation of the function f(x)=log2x after it has been shifted up 5 units transforms to f(x)=log2x+5. The vertical asymptote at x=0 remains the same. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.

x log2x log2x+5

Figure 1. Shifting the graph of f(x)=log2x up 5 units.

18 3 2
14 2 3
12 1 4
1 0 5
2 1 6
4 2 7
8 3 8
Table 1. f(x)=log2x is transformed to f(x)=log2x+5.

If we shift the graph of the function f(x)=log2x down 6 units, all of the points on the graph decrease their y-coordinates by 6, but their x-coordinates remain the same. Therefore, the equation of the function f(x)=log2x after it has been shifted down 6 units transforms to f(x)=log2x6.The vertical asymptote at x=0 remains the same. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.

x log2x log2x6

Figure 2. Shifting the graph of f(x)=log2x down 6 units.

18 3 9
14 2 8
12 1 7
1 0 6
2 1 5
4 2 4
8 3 3
Table 2. f(x)=log2x is transformed to f(x)=log2x6.

Notice that vertical shifts up or down do not change the vertical asymptote.

Move the red dots in manipulation 1 to change the values of b and k. Pay attention to what happens to the graph and the relationship between the value of k and the transformed function.

Manipulation 1. Vertical shifts

Vertical shifts

We can represent a vertical shift of the graph of f(x)=log2x by adding or subtracting a constant, k, to the function:

f(x)=log2x+k

 If k>0, the graph shifts upwards and if k<0 the graph shifts downwards.

Example 1

  1. If f(x)=log3x is shifted vertically up by 7 units, what is the equation of the transformed function?
  2. If f(x)=log7x is shifted vertically down by 4 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5x+9?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=log2x3?

Solution

With vertical shifts, the parent function f(x)=logbx is transformed to f(x)=logbx+k.

  1. k=7 so the transformed function is f(x)=log3x+7
  2. k=4 so the transformed function is f(x)=log7x4
  3. k=9 so the transformation was a vertical shift up by 9 units.
  4. k=3 so the transformation was a vertical shift down by 3 units.

Try It 1

  1. If f(x)=log3x is shifted vertically up by 2 units, what is the equation of the transformed function?
  2. If f(x)=log7x is shifted vertically down by 9 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5x+3?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=log2x8?

Horizontal Shifts

If we shift the graph of the function f(x)=log2x right 8 units, all of the points on the graph increase their x-coordinates by 8, but their y-coordinates remain the same. The x-intercept (1, 0) in the original graph is moved to (9, 0) (figure 3). The vertical asymptote at x=0 shifts right by 8 units to x=8. Any point (x,y) on the original graph is moved to (x+8,y).

But what happens to the original function f(x)=log2x? An automatic assumption may be that since x moves to x+8 that the function will become f(x)=log2(x+8). But that is NOT the case. Remember that the x-intercept is moved to (9, 0) and if we substitute x=9 into the function f(x)=log2(x+8) we get f(9)=log2(9+8)=4.08751!! The way to get a function value of 0 is for the transformed function to be f(x)=log2(x8). Then f(9)=log2(98)=0. So the function f(x)=log2x transforms to f(x)=log2(x8) after being shifted 8 units to the right. The reason is that when we move the function 8 units to the right, the x-value increases by 8 and to keep the corresponding y-coordinate the same in the transformed function, the x-coordinate of the transformed function needs to subtract 8 to get back to the original x that is associated with the original y-value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 3.

x x8 log2(x8)

Figure 3. Shifting the graph right 8 units.

658 18 3
334 14 2
172 12 1
9 1 0
10 2 1
12 4 2
16 8 3
Table 3. Shifting the graph right by 8 units transforms f(x)=log2x into f(x)=log2(x8).

Notice that the vertical asymptote also shifts from x=0 to x=8.

On the other hand, if we shift the graph of the function f(x)=log2x left by 11 units, all of the points on the graph decrease their x-coordinates by 11, but their y-coordinates remain the same. So any point (x,y) on the original graph moves to (x11,y). Consequently, to keep the same y-values we need to increase the x-value by 11 in the transformed function. The equation of the function after being shifted left 11 units is f(x)=log2(x+11). Table 4 shows the changes to specific values of this function, and the graph is shown in figure 4.

x x+11 f(x)=log2(x+11)

Figure 4. Shifting the graph left 11 units.

878 18 –3
434 14 –2
212 12 –1
–10 1 0
–9 2 1
–7 4 2
–3 8 3
Table 4. Shifting the graph left by 11 units transforms f(x)=log2x into f(x)=log2(x+11).

Notice that the vertical asymptote also shifts from x=0 to x=11.

Move the red dots in manipulation 2 to change the values of b and h. Pay attention to what happens to the graph and the relationship between the value of h and the transformed function.


Manipulation 2. Horizontal shifts

horizontal shifts

We can represent a horizontal shift of the graph of f(x)=log2x by adding or subtracting a constant, h, to the variable x.

f(x)=log2(xh)

If h>0 the graph shifts toward the right and if h<0 the graph shifts to the left. The vertical asymptote x=0 shifts to x=h.

Example 2

  1. If f(x)=log3x is shifted right by 2 units, what is the equation of the transformed function?
  2. If f(x)=log7x is shifted left by 9 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5(x+3)?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=log2(x8)?

Solution

With horizontal shifts, the parent function f(x)=logbx is transformed to f(x)=logb(xh).

  1. h=2 so the transformed function is f(x)=log3(x2)
  2. h=9 so the transformed function is f(x)=log7(x+9)
  3. h=3 so the transformation was a horizontal shift left by 3 units.
  4. h=8 so the transformation was a horizontal shift right by 8 units

Try It 2

  1. If f(x)=log3x is shifted right by 7 units, what is the equation of the transformed function?
  2. If f(x)=log7x is shifted left by 4 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5(x5)?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=log2(x+4)?

We can combine vertical and horizontal shifts by transforming f(x)=logb(xh)+k.

Move the red dots in manipulation 3 to change the values of b,h and k. Pay attention to what happens to the graph and the relationship between the values of h andh and the transformed function.


Manipulation 3. Vertical and horizontal shifts

Example 3

  1. If f(x)=log3x is shifted vertically down by 3 units and right by 2 units, what is the equation of the transformed function?
  2. If f(x)=log7x is shifted left by 9 units and up by 6 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5(x2)+7?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=log2(x+5)4?

Solution

With horizontal and vertical shifts, the parent function f(x)=logbx is transformed to f(x)=logb(xh)+k.

  1. h=2,k=3 so the transformed function is f(x)=log3(x2)3
  2. h=9,k=6 so the transformed function is f(x)=log7(x+9)+6
  3. h=2,k=7 so the transformation was a horizontal shift right by 2 units and up by 7 units.
  4. h=5,k=4 so the transformation was a horizontal shift left by 5 units and down by 4 units.

Try It 3

  1. If f(x)=log3x is shifted down by 7 units and right by 5 units, what is the equation of the transformed function?
  2. If f(x)=log7x is shifted left by 4 units and up by 3 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5(x2)1?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=log2(x+4)+7?

Vertical Stretching and Compressing

If we vertically stretch the graph of the function f(x)=log2x by a factor of 2, all of they-coordinates of the points on the graph are multiplied by 2, but their x-coordinates remain the same. The equation of the function after the graph is stretched up by a factor of 2 is f(x)=2log2x. The reason for multiplying  log2x by 2 is that each y-coordinate is doubled, and since y=log2x, log2x is doubled. Table 5 shows this change and the graph is shown in figure 5.

x log2x f(x)=2log2x

Figure 5. Stretching the graph vertically.

18 -3 -6
14 -2 -4
12 -1 -2
1 0 0
2 1 2
4 2 4
8 3 6
Table 5. Stretching the graph vertically by a factor of 2 transforms f(x)=log2x into f(x)=2log2x.

On the other hand, if we vertically compress the graph of the function f(x)=log2x to half of its original height, we multiply the function by the factor 12. All of the y-coordinates of the points on the graph are halved, but their x-coordinates remain the same. This means the y-coordinates are divided by 2, or multiplied by 12. The equation of the function after being compressed is f(x)=12×log2x. The reason for multiplying log2x by 12 is that each y-coordinate becomes half of the original value when it is divided by 2. Table 6 shows this change and the graph is shown in figure 6.

x log2x f(x)=12log2x

Figure 6. Compressing the graph vertically.

18 -3 32
14 -2 -1
12 -1 12
1 0 0
2 1 12
4 2 1
8 3 32
Table 6. Compressing the graph vertically by a factor of 12 transforms f(x)=log2x into f(x)=12log2x.

Notice that vertical stretching and compressing do not change the vertical asymptote.

Move the red dots in manipulation 4 to change the values of b and a. Pay attention to what happens to the graph and the relationship between the value of a and the transformed function.


Manipulation 4. Vertical stretching and compressing

vertical stretching and compressing

A stretch or compression of the graph of f(x)=log2x can be represented by multiplying the function by a constant, a>0.

f(x)=alog2x

The magnitude of a indicates the stretch/compression of the graph. If a>1, the graph is stretched up by a factor of a. If [latex]0

Example 4

  1. If f(x)=log3x is stretched by a factor of 7, what is the equation of the transformed function?
  2. If f(x)=log7x is compressed to one-third its height, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=16log5x?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=5log2x?

Solution

With stretching and compression, the parent function f(x)=logbx is transformed to f(x)=alogbx.

  1. a=7 so the transformed function is f(x)=7log3x
  2. a=13 so the transformed function is f(x)=13log7x
  3. a=16 so the transformation was a compression to one-sixth its height
  4. a=5 so the transformation was a vertical stretch by a factor of 5

Try It 4

  1. If f(x)=log3x is stretched by a factor of 2, what is the equation of the transformed function?
  2. If f(x)=log7x is compressed to one-eighth its height, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=14log5x?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=7log2x?

Now we can combine vertical stretches and compressions with horizontal and vertical shifts.

Example 5

  1. If f(x)=log3x is stretched by a factor of 7, and moved down by 5 units what is the equation of the transformed function?
  2. If f(x)=log7x is compressed to one-third its height, moved left by 4 units and moved up by 2 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=15log5(x9)6?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=3log2(x+5)+7?

Solution

With stretching and compression, combined with shifting the parent function f(x)=logbx is transformed to f(x)=alogb(xh)+k.

  1. a=7,k=5 so the transformed function is f(x)=7log3x5
  2. a=13,h=4,k=2 so the transformed function is f(x)=13log7(x+4)+2
  3. a=15,h=9,k=6 so the transformation was a compression to one-fifth its height, a shift right by 9 units, and a shift down by 6 units
  4. a=3,5,k=7 so the transformation was a vertical stretch by a factor of 3, a shift left by 5 units, and a shift up by 7 units

Try It 5

  1. If f(x)=log3x is stretched by a factor of 5, and moved down by 2 units what is the equation of the transformed function?
  2. If f(x)=log7x is compressed to one-half its height, moved left by 7 units and moved up by 3 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=13log5(x2)3?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=9log2(x+1)+5?

Reflections

Across the x-axis

When the graph of the function f(x)=log2x is reflected across the x-axis, the y-coordinates of all of the points on the graph change their signs, from positive to negative values or from negative to positive values, while the x-coordinates remain the same. The equation of the function after f(x)=log2x is reflected across the x-axis is f(x)=log2x. The graph changes from increasing upwards to decreasing downwards. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 7.

x log2x log2x

Figure 7. Reflecting the graph of f(x)=log2x across the x-axis.

18 -3 3
14 -2 2
12 -1 1
1 0 0
2 1 -1
4 2 -2
8 3 -3
Table 7. Reflecting the graph of f(x)=log2x across the x-axis transforms f(x)=log2x into f(x)=log2x.

Notice that reflecting across the x-axis does not change the vertical asymptote.

Across the y-axis

When the graph of the function f(x)=log2x is reflected across the y-axis, the x-coordinates of all of the points on the graph change their signs, from positive to negative values, while the y-coordinates remain the same. The equation of the function after f(x)=log2x is reflected across the y-axis is f(x)=log2(x). The graph changes from increasing from the left to decreasing from the left. Table 8 shows the effect of such a reflection on the functions values and the graph is shown in figure 8.

x x log2(x)

Figure 8. Reflecting the graph of f(x)=log2x across the y-axis.

18 18 –3
14 14 –2
12 12 –1
–1 1 0
–2 2 1
–4 4 2
–8 8 3
Table 8. Reflecting the graph of f(x)=log2x across the y-axis transforms f(x)=log2x into f(x)=log2(x)
Notice that reflecting across the x-axis does not change the vertical asymptote.

Example 6

  1. If f(x)=log3x is reflected across the x-axis what is the equation of the transformed function?
  2. If f(x)=log7x is reflected across the y-axis what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5x6?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=3log2x?

Solution

  1. The y-values change sign so the transformed function is f(x)=log3x
  2.  The x-values change sign so the transformed function is f(x)=log7(x)
  3. a=1,k=6 so the transformation was a reflection across the x-axis and a shift down by 6 units
  4. a=3 and x is x so the transformation was a vertical stretch by a factor of 3, a reflection across the x, and a reflection across the y-axis

Try It 6

  1. If f(x)=log9x is reflected across the x-axis what is the equation of the transformed function?
  2. If f(x)=log4x is reflected across the y-axis what is the equation of the transformed function?
  3. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5(x)+4?
  4. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=2log2(x)?

Combining Transformations

After learning all the transformations for the function f(x)=logbx, we should be able to write a transformed function given specific transformations, and also determine what transformations have been performed on the function f(x)=logbx, given an arbitrary transformed function f(x)=alogb(xh)+k.

Example 7

What transformations have been done to the parent function f(x)=log2x to get the transformed function f(x)=3log2(x+3)6?

Solution

We need to identify a,h,k and whether or not x has a negative sign in front of it. To do this we line up the transformed function f(x)=3log2(x+3)6 with the standard function f(x)=alog2(xh)+k:

a=3 means it has been stretched by a factor of 3 and reflected across the x-axis.

h=3 means it has been shifted left by 3 units.

k=6 means it has been shifted down by 6 units.

Try It 7

  1. What transformation was made to the parent function f(x)=log5x if the transformed function is f(x)=log5x+7?
  2. What transformation was made to the parent function f(x)=log2x if the transformed function is f(x)=2log2(x5)4?

Example 8

  1. If f(x)=log3x is reflected across the x-axis, stretched by a factor of 3, and shifted left by 2 units, what is the equation of the transformed function? What happens to the vertical asymptote?
  2. If f(x)=log7x is reflected across the y-axis, compressed to half its height, and shifted up by 7 units, what is the equation of the transformed function? What happens to the vertical asymptote?

Solution

  1. a=3,h=2 so the transformed function is 3log3(x+2). The vertical asymptote is shifted left by 2 units from x=0 to x=2.
  2. a=12,k=7 and x has a negative coefficient so the transformed function is f(x)=12log7(x)+7. Since there are no horizontal shifts, nothing happens to the vertical asymptote.

Try It 8

  1. If f(x)=log3x is stretched by a factor of 7, shifted right by 4 units and shifted down by 4 units, what is the equation of the transformed function? What happens to the vertical asymptote?
  2. If f(x)=log7x is reflected across the x-axis, compressed to one-sixth its height, shifted left by 1 unit, and shifted up by 7 units, what is the equation of the transformed function? What happens to the vertical asymptote?

Move the red dots in manipulation 5 to change the values of a,h,k and b or to reflect the graph across the y-axis. Pay attention to what happens to the graph and the relationship between the values of a,h,k and the transformed function.


Manipulation 5. Transformations on f(x)=logbx