# Advanced Geometry : How to graph a logarithm

## Example Questions

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### Example Question #1 : Coordinate Geometry

What is the -intercept of the graph of  ?

Possible Answers:

The graph has no -intercept.

Correct answer:

Explanation:

Set  and solve:

The  -intercept is .

### Example Question #2 : Coordinate Geometry

What is the -intercept of the graph of  ?

Possible Answers:

The graph has no -intercept.

Correct answer:

Explanation:

Set  and evaluate :

Since

, and the -intercept is .

### Example Question #3 : Coordinate Geometry

What is the vertical asymptote of the graph of  ?

Possible Answers:

The graph has no vertical asymptote.

Correct answer:

Explanation:

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 .

### Example Question #4 : Coordinate Geometry

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

The graph of  has no -intercept.

Correct answer:

Explanation:

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

This can be rewritten in exponential form:

The -intercept of the graph of  is .

### Example Question #5 : Coordinate Geometry

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

The graph of  has no -intercept.

Correct answer:

The graph of  has no -intercept.

Explanation:

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.

### Example Question #6 : Coordinate Geometry

Define a function  as follows:

Give the equation of the vertical asymptote of the graph of .

Possible Answers:

Correct answer:

Explanation:

Only positive numbers have logarithms, so

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

### Example Question #7 : Coordinate Geometry

Define a function  as follows:

Give the equation of the vertical asymptote of the graph of .

Possible Answers:

The graph of  has no vertical asymptote.

Correct answer:

Explanation:

Only positive numbers have logarithms, so

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

### Example Question #8 : Coordinate Geometry

Define a function  as follows:

Give the -intercept of the graph of .

Possible Answers:

The graph of  has no -intercept.

Correct answer:

Explanation:

The -coordinate of the -intercept is :

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

.

The -intercept is .

### Example Question #9 : Coordinate Geometry

Define functions  and  as follows:

Give the -coordinate of a point at which the graphs of the functions intersect.

Possible Answers:

The graphs of  and  do not intersect.

Correct answer:

Explanation:

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.

### Example Question #10 : Coordinate Geometry

Let  be the point of intersection of the graphs of these two equations:

Evaluate .

Possible Answers:

Correct answer:

Explanation:

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.

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