Graphing a logarithm - GMAT Quantitative

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.

Set and solve:

The -intercept is .

Set and evaluate :

Since ,

, and the -intercept is .

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 .

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 .

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

Rewrite in exponential form:

.

The -intercept is .

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.

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.

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.

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:

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

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 .

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.

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.

Set and solve:

The -intercept is .