SOLUTION: Question: find the domain, vertical asymptote, and the x-intercept of f(x)=log(subscript 4)(5-x).

Question 201505: Question: find the domain, vertical asymptote, and the x-intercept of f(x)=log(subscript 4)(5-x).
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Domain: The arguments to any logarithm function, no matter what the base, must always be positive. So to find the domain we just have to solve:

Adding x to both sides gives us
which says x is less than 5. (Always read inequalities from where the variable is! In this case we read it from right to left, since x is on the right, which is why it is a less than.)
The domain is all numbers less than 5.

Vertical asymptote: For logarithm functions the vertical asymptote will be for x values that would make the argument zero. So we solve

Again adding x to both sides we get

So the vertical asymptote is the line: x = 5

The x-intercepts of a function is where the y-value is 0. So we need to solve
0 = log(base4)(5-x)
If we understand exponents and logarithms, this is easy. This equation says that 0 (zero) is the exponent you put on 4 to get (5-x). But a zero exponent on any number (except 0) always results on 1! So this equation means that (5-x) must be 1.
Solving

Subtract 5 from both sides giving

Dividing (or multiplying) both sides by -1 we get

So our only x-intercept is (4, 0).

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