We're asked to graph the given log function
as well as give the domain range
and the equation of the vertical asymptote.
We are given y equals log base four of x.
Because we have the basic log function here,
where the log function is in the form
y equals log base b of x,
where by definition, x has to be greater than zero,
we know the domain is going to be
the open interval from zero to infinity
using interval notation.
The range is going to be all real numbers
or the open interval from
negative infinity to positive infinity
and the vertical asymptote is going to be x equals zero.
Again, all this will be true for the given function
because it fits the form of
the basic log function shown here.
And therefore, we will not list the information again.
And now, to graph the function,
we'll write the equivalent expediential equation
and us the expediential equation
to complete the table of values.
So, because we have the log isolated,
we can write the equivalent expediential equation
using the definition shown here below,
where the equivalent expediential equation
to y equals log base four of x
is four raised to the power of y equals x.
So, we'll use the equation
four raised to the power of y equals x
to complete the table of values.
However, instead of selecting x and determining y,
because y is in the position of the exponent,
we will select y and determine x.
I do want to make a note.
In the online homework, you only have to find
the horizontal intercept and one other point.
But for this explanation,
we'll complete the entire table of values.
Let's begin by determining the horizontal intercept,
where any point on the horizontal axis or x-axis
has a y value of zero, we'll begin by letting y equal zero.
So, when y is equal to zero,
x is equal to four raised to the power of zero
and four raised to the power of zero is equal to one.
So, notice how this gives us the ordered pair 1,0,
which is the horizontal intercept.
And now, let's let y equal one.
When y is one, x is equal to four raised to the first power,
giving us x equals four.
Let's try y equals two.
When y is two,
x is equal to four squared and four squared is equal to 16.
So, while this does give us a point on the graph,
notice how 16,2 is not on the given coordinate plane,
so it's not going to help us make a graph.
Let's go back and select a y value,
let's say, between zero and one
as well as between one and two.
Let's try our y value of one half
and a y value of 1.5 or three halves.
So, when y is equal to one half,
we have x equals four raised to the power of one half,
which in this case, comes out very nice.
Remember, four to the one half power
is equal to the square root of four, which is equal to two.
But I do want to make a note that sometimes,
these values won't come out so nice.
You may have to use a calculator
and get a decimal approximation.
And even though you won't have an exact value for x,
the ordered pair still help make an accurate graph.
And now when y is equal to three halves,
we have x equals four raised to the power of three halves.
So, this is a nice review.
Let's write this in medical form,
where the denominator's the index
and the numerator's the exponent.
So, x is equal to the square root of four cubed.
And we know the square root of four is equal to two,
this gives us two cubed, and therefore, x is equal to eight.
So, again, a lot of times, these values won't come out nice.
We'll have to evaluate them on the calculator
and get a rounded value for x.
In this case, it comes out nice.
So now, we can make a nice graph of the function
using these ordered pairs and the vertical asymptote,
which you know is x equal zero.
Let's first guess the vertical asymptote
of x equals zero, which is the y-axis.
And again, in the online homework,
you select the log function graphing tool,
click on the horizontal intercept, this point here,
and then one other point, for example, 4,1.
But let's go ahead and apply
as many of these points as we can.
So, 1,0 is here.
4,1 is here.
16,2 is off the screen.
2,1/2 is here.
And 8,3/2 or 8,1.5 is here.
So, we know the graph passes through these four points
and approaches the vertical asymptote
and therefore, the graph looks something like this.
I hope you found this helpful.