We want to find the domain of the given log functions and then state the domain using
inequalities and interval notation. Looking at our first function, we have y equals log base two
of the quantity two x minus four. The quantity two x minus four is the number part of the log
that's obtained by raising two to the power of y. Remember, in exponential form, two is the base,
y is the exponent, and the number would be the quantity two x minus four.
So if we think about all the positive values we get when raising two to a power,
well, the result is always going to be greater than zero.
That's the reason why to find the domain, we have to solve the inequality two x minus
four is greater than zero. The number part of a logarithm is always going to be greater than zero.
So now we'll add four to both sides, which would give us two x is greater
than four. Now we divide both sides by two, so we have x is greater than two. Therefore, our domain
would be x greater than two.
Using inequalities, if we wanted to express this using interval notation, it might be helpful to
sketch a graph. So if this is two, we'd have an open point on two and an arrow to the right
approaching positive infinity. Therefore, using interval notation, we have the interval from
two to infinity, and it's open on two because it does not include two.
Okay, looking at our second example, notice how there's no base listed on this log.
Therefore, we know it's common log or log base
ten. Again, if we focus on the number part of the log, the quantity x squared minus three x would be
obtained by raising ten to the power of y, which would always be greater than zero. Therefore,
to find the domain of this function, we need to solve the inequality x squared minus three x
greater than zero. The solutions to this will be the domain of our function.
To solve a quadratic inequality, we first find the solutions
as if it was an equation. This does factor, so we want to first solve the equation
x times the quantity x minus three is equal to zero. This would be zero when x equals zero
or when x equals three. So these values of x squared would be equal to zero. We want to
find the x values where it's greater than zero. So what we'll do now is sketch a number line.
Plot these two values as open points because of the inequality symbol.
Then we will test x values in each of these three intervals to see
which satisfy the original inequality. So let's test x equals negative one,
let's say one, and let's say four. If we let x equal four, we'd have four squared
is sixteen minus twelve, which is four, and that is greater than zero, so this interval is true.
Let's go ahead and graph it approaching positive infinity.
When x is one, we'd have one minus three, which is negative two, and that's not
greater than zero, so this is false. When x is negative one, we'd have positive one
minus three times negative one, which becomes one plus three, which is four, and that's greater
than zero, so this is true. This interval is also part of the domain approaching negative infinity.
So the domain using inequalities would be x is less than zero or x is greater than three. When
we are using interval notation, we would have the interval from negative infinity to zero
and the interval from three to infinity. Again,
it does not include these endpoints, therefore we have open intervals on three and zero.
Okay, thank you for watching.