Question

In the following exercises, graph each logarithmic function.

$y={\log }_{\frac{1}{4}}\left(x\right)$

Short answer

The graph of the function $y={\log }_{\frac{1}{4}}\left(x\right)$is

Step-by-step solution

Step 1. Convert into exponential function 

To graph a logarithmic function, it is easiest to convert the equation to its exponential form,

Consider the logarithmic function $y={\log }_{\frac{1}{4}}\left(x\right)$.

The equivalent exponential function is given as role="math" localid="1645082661169" ${\left(\frac{1}{4}\right)}^y=x$.

Step 2. Make a table of values 

We will use the exponential function ${\left(\frac{1}{4}\right)}^y=x$to make a table of values.

$y$	${\left(\frac{1}{4}\right)}^y=x$	$(x,y)$
$-2$	${\left(\frac{1}{4}\right)}^{-2}=16$	$(16,-2)$
$-1$	${\left(\frac{1}{4}\right)}^{-1}=4$	$(4,-1)$
$0$	${\left(\frac{1}{4}\right)}^0=1$	$(1,0)$
$1$	${\left(\frac{1}{4}\right)}^1=\frac{1}{4}$	$(\frac{1}{4},1)$
$2$	${\left(\frac{1}{4}\right)}^2=\frac{1}{16}$	$(\frac{1}{16},2)$

Step 3. Draw the graph

Using the table of values the graph of the function is given as