Graph logarithms

Evaluating logarithms

Recall that a logarithmic equation of the form $\log_ba=c$logb​a=c is equivalent to an exponential equation of the form $b^c=a$bc=a. That is, the value of the logarithm $\log_ba$logb​a 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,

$\log_28=\log_2\left(2^3\right)=3$log2​8=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

$\log_ba=\frac{\log_na}{\log_nb}$logb​a=logn​alogn​b​

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

$\log_213=\frac{\log_{10}13}{\log_{10}2}\approx3.7004$log2​13=log10​13log10​2​≈3.7004.

 

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

Here is a graph of the function $y=\log_2x$y=log2​x:

 

 

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=logb​x, 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=alogb​x 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=−5log2​x:

 

 

• Adding a constant to get $y=\log_bx+k$y=logb​x+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=log2​x+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):

 

 

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=alogb​x still intercepts the $x$x-axis at the point $\left(1,0\right)$(1,0).

 

 

Practice questions

QUESTION 1

QUESTION 2

QUESTION 3