In this video,
we will graph the given log function by hand,
as well as give the domain, range,
and the equation of the vertical asymptote.
Remember, for the basic log function
y equals log base b of x, x must be greater than zero.
And now looking at our function,
we have log base three of x plus one.
So for this function, x plus one must be greater than zero.
So if we solve this inequality, we can determine the domain.
If we subtract one on both sides,
we have x is greater than negative one,
which is the domain.
Using interval notation is the open interval
from negative one to infinity.
And now the zero of x plus one
or the solution to x plus one equals zero
will give us the equation of the vertical asymptote,
which is x equals negative one.
And now for the basic log function,
the output or range is the open interval
from negative infinity to positive infinity
or all real numbers.
So if the output of this function is all real numbers,
multiplying by two and then subtracting three
will not change the range.
The range is also the open interval
from negative infinity to positive infinity.
And now to graph this by hand,
we will first isolate the logarithm,
write the equivalent exponential equation,
and then use the exponential equation
to complete the table of values.
To isolate the logarithm,
we first add three to both sides,
which gives us y plus three equals
two times log base three of x plus one.
And now we divide both sides by two,
giving us the quantity y plus three divided by two
is equal to log base three of the quantity x plus one.
Now that we have the log by itself,
we now write the equivalent exponential equation
using the definition below, which is three
raised to the power of the quantity y plus three
divided by two must equal x plus one.
So again, the exponential equation is three
raised to the power of y plus three
divided by two equals x plus one.
Let's solve this for x, though,
before we complete the table.
To solve for x, we subtract one on both sides,
which gives us the equation three
raised to the power of y plus three
divided by two minus one equals x.
We will use this form
of the exponential equation to complete the table.
However, instead of selecting x and determining y,
we will select y and determine x.
Notice how, if y is equal to negative three,
we would have a zero exponent here.
So let's begin with y equals negative three.
So again, when y is negative three,
we have x equals,
the exponent would be negative three plus three,
which is zero.
Zero divided by two is still zero,
giving us three to the zero minus one.
Well, three to the zero is one.
One minus one is zero.
So we know the ordered pair zero comma negative three
would be one point on the graph of the function.
And now looking at the exponent,
notice how we are dividing y plus three by two,
so we do want y plus three to be even.
So if we increase negative three by two,
the sum will still be even.
Let's let y equal negative one.
So when y is negative one,
we have x equals three raised to the power of,
negative one plus three is two,
two divided by two is one,
giving us three to the first minus one,
giving us three minus one, or two.
And now let's increase y by two again.
Let's try y equals positive one.
When y is positive one, we have x equals three
raised to the power of one plus three.
That's four.
Four divided by two is two
and then minus one.
Three squared is nine.
Nine minus one is eight, giving us x equals eight.
Let's see if this is enough information
to make a nice graph of the function.
Let's first sketch the vertical asymptote
of x equals negative one, which is this vertical line here.
And now let's plot these three points.
We have zero comma negative three.
We have two comma negative one.
And we have eight comma one,
which is right here on the edge of the coordinate plane.
So we know the graph passes through these three points
and approaches the vertical asymptote,
and therefore the graph looks something like this.
I hope you found this helpful.