The graph of a logarithmic function has a vertical asymptote which can be found by finding the value at which the power is equal to 0:

If , then  is an undefined expression, so the vertical asymptote is .

Set  and evaluate  to find the -coordinate of the -intercept.

This can be rewritten in exponential form:

The -intercept of the graph of  is .

The -coordinate of the -intercept is :

However, the logarithm of a negative number is an undefined expression, so  is an undefined quantity, and the graph of  has no -intercept.

Only positive numbers have logarithms, so

The graph never crosses the vertical line of the equation , so this is the vertical asymptote. 

Only positive numbers have logarithms, so

The graph never crosses the vertical line of the equation , so this is the vertical asymptote. 

The -coordinate of the -intercept is :

Since 2 is the cube root of 8, , and  . Therefore, 

.

The -intercept is .

Since , the definition of  can be rewritten as follows:

First, we need to find the -coordinate of the point at which the graphs of  and  meet by setting 

Since the common logarithms of the polynomial and the rational expression are equal, we can set those expressions themselves equal, then solve:

We can solve using the  method, finding two integers whose sum is 24 and whose product is  - these integers are 10 and 14, so we split the niddle term, group, and factor: 

or 

This gives us two possible -coordinates. However, since 

,

an undefined quantity - negative numbers not having logarithms -

we throw this value out. As for the other -value, we evaluate: 

and 

 is the correct -value, and  is the correct -value.

Substitute  and  for  and , respectively, and solve the resulting system of linear equations:

Multiply the first equation by 2, and the second by 3, on both sides, then add:

Now back-solve:

We need to find both  and  to ensure a solution exists. By substituting back:

.

 is the solution, and , the correct choice.

Substitute  and  for  and , respectively, and solve the resulting system of linear equations:

Multiply the first equation by 2, and the second by 3, on both sides, then add:

Back-solve:

We need to find both  and  to ensure a solution exists. By substituting back:

and

We check this solution in both equations:

 - true.

 - true.

 is the solution, and , the correct choice.

Given the function , the vertical asymptote can be found by observing that a logarithm cannot be taken of a number that is not positive. Therefore, it must hold that , or, equivalently,  and that the graph of  will never cross the vertical line . That makes  the vertical asymptote, so it follows that the graph with vertical asymptote  will have  in the  position. The only choice that meets this criterion is