NZ Level 7 (NZC) Level 2 (NCEA)
Logarithm Graph
Lesson

### Evaluating logarithms

Recall that a logarithmic equation of the form $\log_ba=c$logba=c is equivalent to an exponential equation of the form $b^c=a$bc=a. That is, the value of the logarithm $\log_ba$logba is "the power which $b$b must be raised to in order to give $a$a".

Using this, if $a$a is a rational power of $b$b then we can simplify the logarithm nicely. For example,

$\log_28=\log_2\left(2^3\right)=3$log28=log2(23)=3.

In any other case, where $a$a is not a rational power of $b$b, we can use a calculator (or other technology) to approximate the value. In particular, we can use the logarithm property

$\log_ba=\frac{\log_na}{\log_nb}$logba=lognalognb

to rewrite any logarithm using a base that is available on the calculator. For example,

$\log_213=\frac{\log_{10}13}{\log_{10}2}\approx3.7004$log213=log1013log1023.7004.

### Graphs of the form $y=\log_bx$y=logb​x

Here is a graph of the function $y=\log_2x$y=log2x:

The function $y=\log_2x$y=log2x.

From the graph, we can see a few features:

• The graph only exists for positive values of $x$x, with an asymptote along the $y$y-axis (the line $x=0$x=0).
• The curve intersects the $x$x-axis at the point $\left(1,0\right)$(1,0).
• The function is increasing, but at a decreasing rate.

In fact, these features are true for any function of the form $y=\log_bx$y=logbx, where $b>1$b>1.

### Graphs of the form $y=a\log_bx+k$y=alogb​x+k

As with other functions that we've looked at, we can transform the graphs of logarithms in a few ways:

• Multiplying by a constant to get $y=a\log_bx$y=alogbx corresponds to vertically scaling the graph by a factor of $a$a. This can also involve a reflection about the $x$x-axis if $a$a is negative. For example, here is a graph of $y=-5\log_2x$y=5log2x:

A graph of the function $y=-5\log_2x$y=5log2x.

• Adding a constant to get $y=\log_bx+k$y=logbx+k corresponds to vertically translating the graph by $k$k units. This translation is up if $k$k is positive and down if $k$k is negative. For example, here is a graph of $y=\log_2x+5$y=log2x+5:

A graph of the function $y=\log_2x+5$y=log2x+5

• We can also obtain the graph of $y=\log_b\left(-x\right)$y=logb(x) which corresponds to a reflection about the $y$y-axis. For example, here is a graph of $y=\log_2\left(-x\right)$y=log2(x):

A graph of the function $y=\log_2\left(-x\right)$y=log2(x).

Note that in all of these cases, the asymptote of the function is still along the $y$y-axis. Additionally, a function of the form $y=a\log_bx$y=alogbx still intercepts the $x$x-axis at the point $\left(1,0\right)$(1,0).

Graphs of the form $y=a\log_bx+k$y=alogbx+k

To sketch the graph of a function of the form $y=a\log_bx+k$y=alogbx+k, start by identifying the locations of the asymptote and the $x$x-intercept.

Then plot a couple of extra points to see the general shape of the graph.

#### Practice questions

##### QUESTION 1

Consider the function $y=3\log_2x$y=3log2x.

1. Solve for the $x$x-coordinate of the $x$x-intercept.

2. Complete the table of values for $y=3\log_2x$y=3log2x.

 $x$x $y$y $\frac{1}{2}$12​ $1$1 $2$2 $4$4 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. State the equation of the vertical asymptote.

4. Sketch the graph of $y=3\log_2x$y=3log2x.

##### QUESTION 2

Consider the function $y=\log_3x-1$y=log3x1.

1. Solve for the $x$x-coordinate of the $x$x-intercept.

2. Complete the table of values for $y=\log_3x-1$y=log3x1.

 $x$x $y$y $\frac{1}{3}$13​ $1$1 $3$3 $9$9 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. State the equation of the vertical asymptote.

4. Sketch the graph of $y=\log_3x-1$y=log3x1.

##### QUESTION 3

Consider the function $y=3\log_4\left(-x\right)$y=3log4(x).

1. Solve for the $x$x-coordinate of the $x$x-intercept.

2. Complete the table of values for $y=3\log_4\left(-x\right)$y=3log4(x).

 $x$x $y$y $-16$−16 $-4$−4 $-1$−1 $-\frac{1}{4}$−14​ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
3. State the equation of the vertical asymptote.

4. Sketch the graph of $y=3\log_4\left(-x\right)$y=3log4(x).

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems