document.write( "Question 273500: What is the domain, range, x-intercept and vertical asymptote of the function:
\n" ); document.write( " log[base 3] (x-4) \n" ); document.write( "

Algebra.Com's Answer #199818 by jsmallt9(3758) $\"\"$ $\"About$

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I'm assuming the function is
\n" ); document.write( " $\"y+=+log%283%2C+%28x-4%29%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"f%28x%29+=+log%283%2C+%28x-4%29%29\"$

\n" ); document.write( "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:

Zeros in denominators. Dividing be zero is undefined.
Zero or negative bases or arguments of logarithms.
Negative radicands (the number inside a radical) of even-numbered roots (like square roots). We cannot allow something like $\"sqrt%28-4%29\"$ to happen because it is not possible to raise any real number to an even power and get a negative result.

\n" ); document.write( "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:
\n" ); document.write( " $\"x-4+%3E+0\"$
\n" ); document.write( "or
\n" ); document.write( " $\"x+%3E+4\"$
\n" ); document.write( "This is our domain.

\n" ); document.write( "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:
\n" ); document.write( "x = 4

\n" ); document.write( "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.

\n" ); document.write( "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:
\n" ); document.write( " $\"0+=+log%283%2C+%28x-4%29%29\"$
\n" ); document.write( "To solve this we rewrite it in exponential form:
\n" ); document.write( " $\"3%5E0+=+x-4\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"1+=+x-4\"$
\n" ); document.write( "Adding 4 to each side we get:
\n" ); document.write( " $\"5+=+x\"$
\n" ); document.write( "So there is one x-intercept: (5, 0)

\n" ); document.write( " \n" ); document.write( "

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