Victorian Curriculum Year 10A - 2020 Edition 5.06 Logarithmic graphs and scales
Lesson

Graphs of logarithmic equations of the form $y=a\log_B\left(x-h\right)+k$y=alogB(xh)+k (where $a$a, $B$B, and $h$h and $k$k are any number and $B>0$B>0) are called logarithmic graphs. The logarithmic graph defined by $y=\log_2x$y=log2x

### Features of logarithmic graphs

Like lines, logarithmic graphs will always have an $x$x-intercept. This is the point on the graph which touches the $x$x-axis. We can find this by setting $y=0$y=0 and finding the value of $x$x. For example, the $x$x-intercept of $y=\log_2x$y=log2x is $\left(1,0\right)$(1,0)

Similarly, we can look for $y$y-intercepts by setting $x=0$x=0 and then solving for $y$y. Because this is an logarithmic equation, there could be $0$0 or $1$1 solutions, and there will be the same number of $y$y-intercepts. For example, the graph of $y=\log_2x$y=log2x has no $y$y-intercept.

Logarithmic graphs have a vertical asymptote which is the vertical line which the graph approaches but does not touch. For example, the vertical asymptote of $y=\log_2x$y=log2x is $x=0$x=0

### Transformations of logarithmic graphs

A logarithmic graph can be vertically translated by increasing or decreasing the $y$y-values by a constant number. So to translate $y=\log_2x$y=log2x up by $k$k units gives us $y=\log_2x+k$y=log2x+k. Vertically translating up by $2$2 ($y=\log_2x+2$y=log2x+2) and down by $2$2 ($y=\log_2x-2$y=log2x2)

Similarly, a logarithmic graph can be horizontally translated by increasing or decreasing the $x$x-values by a constant number. However, the $x$x-value together with the translation must both be in the logarithm. That is, to translate $y=\log_2x$y=log2x to the left by $h$h units we get $y=\log_2\left(x+h\right)$y=log2(x+h). Horizontally translating left by $2$2 ($y=\log_2\left(x+2\right)$y=log2(x+2)) and right by $2$2 ($y=\log_2\left(x-2\right)$y=log2(x2))

A logarithmic graph can be vertically scaled by multiplying every $y$y-value by a constant number. So to expand the logarithmic graph $y=\log_2x$y=log2x by a scale factor of $a$a we get $y=a\log_2x$y=alog2x. We can compress a logarithmic graph by dividing by the scale factor instead. Vertically expanding by a scale factor of $2$2 ($y=2\log_2x$y=2log2x) and compressing by a scale factor of $2$2 ($y=\frac{1}{2}\log_2x$y=12log2x).

We can vertically reflect a logarithmic graph about the $x$x-axis by taking the negative of the $y$y-values. So to reflect $y=\log_2x$y=log2x about the $x$x-axis gives us $y=-\log_2x$y=log2x.

We can similarly horizontally reflect a logarithmic graph about the $y$y-axis by taking the negative of the $x$x-values. So to reflect $y=\log_2x$y=log2x about the $y$y-axis gives us $y=\log_2\left(-x\right)$y=log2(x). Reflecting the logarithmic graph about the $y$y-axis ($y=-\log_2x$y=log2x) and about the $x$x-axis ($y=\log_2\left(-x\right)$y=log2(x))

Summary

The graph of a logarithmic equation of the form $y=a\log_B\left(x-h\right)+k$y=alogB(xh)+k is a logarithmic graph.

Logarithmic graphs have an $x$x-intercept and can have $0$0 or $1$1$y$y-intercepts, depending on the solutions to the logarithmic equation.

Logarithmic graphs have a vertical asymptote which is the vertical line that the graph approaches but does not intersect.

Logarithmic graphs can be transformed in the following ways (starting with the logarithmic graph defined by $y=\log_2x$y=log2x):

• Vertically translated by $k$k units: $y=\log_2+k$y=log2+k
• Horizontally translated by $h$h units: $y=\log_2\left(x-h\right)$y=log2(xh)
• Vertically scaled by a scale factor of $a$a: $y=a\log_2x$y=alog2x
• Vertically reflected about the $x$x-axis: $y=-\log_2x$y=log2x
• Horizontally reflected about the $y$y-axis: $y=\log_2\left(-x\right)$y=log2(x)

#### Practice questions

##### Question 1

Consider the function $y=\log_4x$y=log4x.

1. Complete the table of values for $y=\log_4x$y=log4x, rounding any necessary values to 2 decimal places.

 $x$x $y$y $0.3$0.3 $1$1 $2$2 $3$3 $4$4 $5$5 $10$10 $20$20 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Which of the following is the graph of $y=\log_4x$y=log4x?

A

B

C

D

A

B

C

D
##### Question 2

A graph of the function $y=\log_3x$y=log3x is shown below.

A graph of the function $y=\log_3x+3$y=log3x+3 can be obtained from the original graph by transforming it in some way.

1. Complete the table of values below for $y=\log_3x$y=log3x:

 $x$x $\frac{1}{3}$13​ $1$1 $3$3 $9$9 $\log_3x$log3​x $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

2. Now complete the table of values below for $y=\log_3x+3$y=log3x+3:

 $x$x $\frac{1}{3}$13​ $1$1 $3$3 $9$9 $\log_3x+3$log3​x+3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

3. Which of the following is a graph of $y=\log_3x+3$y=log3x+3?

A

B

C

D

A

B

C

D
4. Which features of the graph are unchanged after it has been translated $3$3 units upwards?

Select all that apply.

The range.

A

The general shape of the graph.

B

The vertical asymptote.

C

The $x$x-intercept.

D

The range.

A

The general shape of the graph.

B

The vertical asymptote.

C

The $x$x-intercept.

D
##### Question 3

Given the graph of $y=\log_6\left(-x\right)$y=log6(x), draw the graph of $y=5\log_6\left(-x\right)$y=5log6(x) on the same plane.