# GMAT Math : Graphing a logarithm

## Example Questions

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### Example Question #71 : Graphing

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

Evaluate .

The system has no solution.

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:

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.

### Example Question #71 : Graphing

The graph of function  has vertical asymptote . Which of the following could give a definition of  ?

Explanation:

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

### Example Question #11 : Graphing

The graph of a function  has -intercept . Which of the following could be the definition of  ?

All of the other choices are correct.

All of the other choices are correct.

Explanation:

All of the functions are of the form . To find the -intercept of such a function, we can set  and solve for :

Since we are looking for a function whose graph has -intercept , the equation here becomes , and we can examine each of the functions by finding the value of .

:

All four choices fit the criterion.

### Example Question #1061 : Gmat Quantitative Reasoning

The graph of a function  has -intercept . Which of the following could be the definition of  ?

Explanation:

All of the functions take the form

for some integer . To find the choice that has -intercept , set  and , and solve for :

In exponential form:

The correct choice is .

### Example Question #15 : Graphing

Define a function  as follows:

Give the -intercept of the graph of .

Explanation:

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

Rewrite in exponential form:

.

The -intercept is .

### Example Question #16 : Graphing

Define functions  and  as follows:

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

The graphs of  and  do not intersect.

The graphs of  and  do not intersect.

Explanation:

Since , the definition of  can be rewritten as follows:

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 two polynomials are equal, we can set the polynomials themselves equal, then solve:

However, if we evaluate , the expression becomes

,

which is undefined, since a negative number cannot have a logarithm.

Consequently, the two graphs do not intersect.

### Example Question #17 : Graphing

The graph of a function  has -intercept . Which of the following could be the definition of  ?

None of the other responses gives a correct answer.

Explanation:

All of the functions are of the form . To find the -intercept of a function ,  we can set  and solve for :

.

Since we are looking for a function whose graph has -intercept , the equation here becomes , and we can examine each of the functions by finding the value of  and seeing which case yields this result.

:

:

:

:

The graph of  has -intercept  and is the correct choice.

### Example Question #18 : Graphing

Define a function  as follows:

A line passes through the - and -intercepts of the graph of . Give the equation of the line.

Explanation:

The -intercept of the graph of  can befound by setting  and solving for :

Rewritten in exponential form:

The -intercept of the graph of  is .

The -intercept of the graph of  can be found by evaluating

The -intercept of the graph of  is .

If  and  are the - and -intercepts, respectively, of a line, the slope of the line is . Substituting  and , this is

.

Setting  and  in the slope-intercept form of the equation of a line:

### Example Question #19 : Graphing

Define functions  and  as follows:

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

The graphs of  and  do not intersect.

Explanation:

Since , the definition of  can be rewritten as follows:

Find the -coordinate of the point at which the graphs of  and  meet by setting

Since the common logarithms of the two polynomials are equal, we can set the polynomials themselves equal, then solve:

The quadradic trinomial can be "reverse-FOILed" by noting that 2 and 6 have  product 12 and sum 8:

Either , in which case

or

, in which case

Note, however, that we can eliminate  as a possible -value, since

,

an undefined quantity since negative numbers do not have logarithms.

Since

and

,

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

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