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`=`a``l``o``g``B`(`x`−`h`)+`k` (where $a$`a`, $B$`B`, and $h$`h` and $k$`k` are any number and $B>0$`B`>0) are called 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`=`l``o``g`2`x` 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`=`l``o``g`2`x` 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`=`l``o``g`2`x` is $x=0$`x`=0

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`=`l``o``g`2`x` up by $k$`k` units gives us $y=\log_2x+k$`y`=`l``o``g`2`x`+`k`.

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`=`l``o``g`2`x` to the left by $h$`h` units we get $y=\log_2\left(x+h\right)$`y`=`l``o``g`2(`x`+`h`).

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`=`l``o``g`2`x` by a scale factor of $a$`a` we get $y=a\log_2x$`y`=`a``l``o``g`2`x`. We can compress a logarithmic graph by dividing by the scale factor instead.

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`=`l``o``g`2`x` about the $x$`x`-axis gives us $y=-\log_2x$`y`=−`l``o``g`2`x`.

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`=`l``o``g`2`x` about the $y$`y`-axis gives us $y=\log_2\left(-x\right)$`y`=`l``o``g`2(−`x`).

Summary

The graph of a logarithmic equation of the form $y=a\log_B\left(x-h\right)+k$`y`=`a``l``o``g``B`(`x`−`h`)+`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`=`l``o``g`2`x`):

- Vertically translated by $k$
`k`units: $y=\log_2+k$`y`=`l``o``g`2+`k` - Horizontally translated by $h$
`h`units: $y=\log_2\left(x-h\right)$`y`=`l``o``g`2(`x`−`h`) - Vertically scaled by a scale factor of $a$
`a`: $y=a\log_2x$`y`=`a``l``o``g`2`x` - Vertically reflected about the $x$
`x`-axis: $y=-\log_2x$`y`=−`l``o``g`2`x` - Horizontally reflected about the $y$
`y`-axis: $y=\log_2\left(-x\right)$`y`=`l``o``g`2(−`x`)

Consider the function $y=\log_4x$`y`=`l``o``g`4`x`.

Complete the table of values for $y=\log_4x$

`y`=`l``o``g`4`x`, rounding any necessary values to 2 decimal places.$x$ `x`$0.3$0.3 $1$1 $2$2 $3$3 $4$4 $5$5 $10$10 $20$20 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Which of the following is the graph of $y=\log_4x$

`y`=`l``o``g`4`x`?Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

A graph of the function $y=\log_3x$`y`=`l``o``g`3`x` is shown below.

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

Loading Graph...

Complete the table of values below for $y=\log_3x$

`y`=`l``o``g`3`x`:$x$ `x`$\frac{1}{3}$13 $1$1 $3$3 $9$9 $\log_3x$ `l``o``g`3`x`$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Now complete the table of values below for $y=\log_3x+3$

`y`=`l``o``g`3`x`+3:$x$ `x`$\frac{1}{3}$13 $1$1 $3$3 $9$9 $\log_3x+3$ `l``o``g`3`x`+3$\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Which of the following is a graph of $y=\log_3x+3$

`y`=`l``o``g`3`x`+3?Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DWhich features of the graph are unchanged after it has been translated $3$3 units upwards?

Select all that apply.

The range.

AThe general shape of the graph.

BThe vertical asymptote.

CThe $x$

`x`-intercept.DThe range.

AThe general shape of the graph.

BThe vertical asymptote.

CThe $x$

`x`-intercept.D

Given the graph of $y=\log_6\left(-x\right)$`y`=`l``o``g`6(−`x`), draw the graph of $y=5\log_6\left(-x\right)$`y`=5`l``o``g`6(−`x`) on the same plane.

- Loading Graph...

Use the definition of a logarithm to establish and apply the laws of logarithms and investigate logarithmic scales in measurement