Question 273500
I'm assuming the function is
{{{y = log(3, (x-4))}}}
or 
{{{f(x) = log(3, (x-4))}}}<br>
The domain is the set of possible values for x. Normally we assume that x can be any real number. But we have to ensure that the various taboos in Math are not violated. What taboos are there in Math? Here are a few of the most well-known:<ul><li>Zeros in denominators. Dividing be zero is undefined.</li><li>Zero or negative bases or arguments of logarithms.</li><li>Negative radicands (the number inside a radical) of even-numbered roots (like square roots). We cannot allow something like {{{sqrt(-4)}}} to happen because it is not possible to raise any real number to an even power and get a negative result.</li></ul>
Your function has no denominators or even-numbered roots. But it does have a logarithm. We must make sure that its base and argument are always positive. The base is 3 which is positive. The argument is x-4. To make the argument positive we need:
{{{x-4 > 0}}}
or
{{{x > 4}}}
This is our domain.<br>
Vertical asymptote. For logarithmic functions the vertical asymptote will be at the "edge" of the domain. To find it, just make an equation out of the domain:
x = 4<br>
Range. Range is the set of possible y values. In this function your y value is a logarithm. Logarithms are exponents and exponents can be any number. So the range is all real numbers.<br>
X-intercept. The x-intercept(s) of a graph are the points where the graph crosses/touches/intersects the x-axis. Since all the points on the x-axis have y coordinates of 0, we find x-intercepts by setting y to zero and solving for x:
{{{0 = log(3, (x-4))}}}
To solve this we rewrite it in exponential form:
{{{3^0 = x-4}}}
which simplifies to:
{{{1 = x-4}}}
Adding 4 to each side we get:
{{{5 = x}}}
So there is one x-intercept: (5, 0)<br>