DSQ: Graphing a logarithm - GMAT Quantitative
Card 1 of 72
Define a function 
as follows:
for nonzero real numbers 
.
Give the equation of the vertical asymptote of the graph of 
.
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 alone gives that 
. 
is the reciprocal of this, or 
, and 
, so the vertical asymptote is 
.
Statement 2 alone gives no clue about either 
, 
, or their relationship.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 alone gives that
Statement 2 alone gives no clue about either
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Give the equation of the vertical asymptote of the graph of 
.
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Each of Statement 1 and Statement 2 gives us only one of 
and 
. However, the two together tell us that
making the vertical asymptote
.
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
Each of Statement 1 and Statement 2 gives us only one of
making the vertical asymptote
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Does the graph of 
have a 
-intercept?
Statement 1: 
.
Statement 2: 
.
Define a function
for nonzero real numbers
Does the graph of
Statement 1:
Statement 2:
Tap to reveal answer
The 
-intercept of the graph of the function 
, if there is one, occurs at the point with 
-coordinate 0. Therefore, we find 
:
This expression is defined if and only if 
is a positive value. Statement 1 gives 
as positive, so it follows that the graph indeed has a 
-intercept. Statement 2, which only gives 
, is irrelevant.
The
This expression is defined if and only if
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Does the graph of 
have a 
-intercept?
Statement 1: 
.
Statement 2: 
and 
have different signs.
Define a function
for nonzero real numbers
Does the graph of
Statement 1:
Statement 2:
Tap to reveal answer
The 
-intercept of the graph of the function 
, if there is one, occurs at the point with 
-coordinate 0. Therefore, we find 
:
This expression is defined if and only if 
is a positive value. However, the two statements together do not give this information; the values of 
and 
from Statement 1 are irrelevant, and Statement 2 does not reveal which of 
and 
is positive and which is negative.
The
This expression is defined if and only if
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
and 
are both positive.
Statement 2: 
and 
are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since 
and 
are of opposite sign, their quotient 
is negative, and 
is positive. This locates the vertical asymptote on the right side of the 
-axis.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
and 
are both positive.
Statement 2: 
and 
are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms, the expression 
must be positive, so
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find the sign of 
; if it is negative, it is on the left side, if it is positive, it is on the right side.
Assume both statements are true. By Statement 1, 
is positive. If 
is positive, then 
is negative, and vice versa. However, Statement 2, which mentions 
, does not give its actual sign - just the fact that its sign is the opposite of that of 
, which we are not given either. The two statements therefore give insufficient information.
Since only positive numbers have logarithms, the expression
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Assume both statements are true. By Statement 1,
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Give the equation of the vertical asymptote of the graph of 
.
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Assume both statements to be true. We need two numbers 
and 
whose sum is 7 and whose product is 12; by trial and error, we can find these numbers to be 3 and 4. However, without further information, we have no way of determining which of 
and 
is 3 and which is 4, so the asymptote can be either 
or 
.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
Assume both statements to be true. We need two numbers
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
What is the equation of the vertical asymptote of the graph of 
?
Statement 1: 
and 
are of opposite sign.
Statement 2: 
Define a function
for nonzero real numbers
What is the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find the sign of 
; if it is negative, it is on the left side, and if it is positive, it is on the right side.
Statement 1 alone only gives us that 
is a different sign from 
; without any information about the sign of 
, we cannot answer the question.
Statement 2 alone gives us that 
, and, consequently, 
. This means that 
and 
are of opposite sign. But again, with no information about the sign of 
, we cannot answer the question.
Assume both statements to be true. Since, from the two statements, both 
and 
are of the opposite sign from 
, 
and 
are of the same sign. Their quotient 
is positive, and 
is negative, so the vertical asymptote 
is to the left of the 
-axis.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Statement 1 alone only gives us that
Statement 2 alone gives us that
Assume both statements to be true. Since, from the two statements, both
← Didn't Know|Knew It →
Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find out whether the signs of 
and 
are the same or different. If 
and 
are of the same sign, then their quotient 
is positive, and 
is negative, putting 
on the left side of the 
-axis. If 
and 
are of different sign, then their quotient 
is negative, and 
is positive, putting 
on the right side of the 
-axis.
Statement 1 alone does not give us enough information to determine whether 
and 
have different signs. 
, for example, but 
, also.
From Statement 2, since the product of 
and 
is negative, they must be of different sign. Therefore, 
is positive, and 
falls to the right of the 
-axis.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Statement 1 alone does not give us enough information to determine whether
From Statement 2, since the product of
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Give the equation of the vertical asymptote of the graph of .
Statement 1:
Statement 2:
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 alone gives that . is the reciprocal of this, or , and , so the vertical asymptote is .
Statement 2 alone gives no clue about either , , or their relationship.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 alone gives that
Statement 2 alone gives no clue about either
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Give the equation of the vertical asymptote of the graph of .
Statement 1:
Statement 2:
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Each of Statement 1 and Statement 2 gives us only one of and . However, the two together tell us that
making the vertical asymptote
.
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
Each of Statement 1 and Statement 2 gives us only one of
making the vertical asymptote
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Does the graph of have a -intercept?
Statement 1: .
Statement 2: .
Define a function
for nonzero real numbers
Does the graph of
Statement 1:
Statement 2:
Tap to reveal answer
The -intercept of the graph of the function , if there is one, occurs at the point with -coordinate 0. Therefore, we find :
This expression is defined if and only if is a positive value. Statement 1 gives as positive, so it follows that the graph indeed has a -intercept. Statement 2, which only gives , is irrelevant.
The
This expression is defined if and only if
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Does the graph of have a -intercept?
Statement 1: .
Statement 2: and have different signs.
Define a function
for nonzero real numbers
Does the graph of
Statement 1:
Statement 2:
Tap to reveal answer
The -intercept of the graph of the function , if there is one, occurs at the point with -coordinate 0. Therefore, we find :
This expression is defined if and only if is a positive value. However, the two statements together do not give this information; the values of and from Statement 1 are irrelevant, and Statement 2 does not reveal which of and is positive and which is negative.
The
This expression is defined if and only if
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Where is the vertical asymptote of the graph of in relation to the -axis - is it to the left of it, to the right of it, or on it?
Statement 1: and are both positive.
Statement 2: and are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since and are of opposite sign, their quotient is negative, and is positive. This locates the vertical asymptote on the right side of the -axis.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 gives irrelevant information. But Statement 2 alone gives sufficient information; since
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Where is the vertical asymptote of the graph of in relation to the -axis - is it to the left of it, to the right of it, or on it?
Statement 1: and are both positive.
Statement 2: and are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms, the expression must be positive, so
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the -axis the vertical asymptote falls, it is necessary to find the sign of ; if it is negative, it is on the left side, if it is positive, it is on the right side.
Assume both statements are true. By Statement 1, is positive. If is positive, then is negative, and vice versa. However, Statement 2, which mentions , does not give its actual sign - just the fact that its sign is the opposite of that of , which we are not given either. The two statements therefore give insufficient information.
Since only positive numbers have logarithms, the expression
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Assume both statements are true. By Statement 1,
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Give the equation of the vertical asymptote of the graph of .
Statement 1:
Statement 2:
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Assume both statements to be true. We need two numbers and whose sum is 7 and whose product is 12; by trial and error, we can find these numbers to be 3 and 4. However, without further information, we have no way of determining which of and is 3 and which is 4, so the asymptote can be either or .
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
Assume both statements to be true. We need two numbers
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
What is the equation of the vertical asymptote of the graph of ?
Statement 1: and are of opposite sign.
Statement 2:
Define a function
for nonzero real numbers
What is the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the -axis the vertical asymptote falls, it is necessary to find the sign of ; if it is negative, it is on the left side, and if it is positive, it is on the right side.
Statement 1 alone only gives us that is a different sign from ; without any information about the sign of , we cannot answer the question.
Statement 2 alone gives us that , and, consequently, . This means that and are of opposite sign. But again, with no information about the sign of , we cannot answer the question.
Assume both statements to be true. Since, from the two statements, both and are of the opposite sign from , and are of the same sign. Their quotient is positive, and is negative, so the vertical asymptote is to the left of the -axis.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Statement 1 alone only gives us that
Statement 2 alone gives us that
Assume both statements to be true. Since, from the two statements, both
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Where is the vertical asymptote of the graph of in relation to the -axis - is it to the left of it, to the right of it, or on it?
Statement 1:
Statement 2:
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the -axis the vertical asymptote falls, it is necessary to find out whether the signs of and are the same or different. If and are of the same sign, then their quotient is positive, and is negative, putting on the left side of the -axis. If and are of different sign, then their quotient is negative, and is positive, putting on the right side of the -axis.
Statement 1 alone does not give us enough information to determine whether and have different signs. , for example, but , also.
From Statement 2, since the product of and is negative, they must be of different sign. Therefore, is positive, and falls to the right of the -axis.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Statement 1 alone does not give us enough information to determine whether
From Statement 2, since the product of
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Give the equation of the vertical asymptote of the graph of .
Statement 1:
Statement 2:
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
Statement 1 alone gives that . is the reciprocal of this, or , and , so the vertical asymptote is .
Statement 2 alone gives no clue about either , , or their relationship.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
Statement 1 alone gives that
Statement 2 alone gives no clue about either
← Didn't Know|Knew It →
Define a function as follows:
for nonzero real numbers .
Give the equation of the vertical asymptote of the graph of .
Statement 1:
Statement 2:
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Tap to reveal answer
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Each of Statement 1 and Statement 2 gives us only one of and . However, the two together tell us that
making the vertical asymptote
.
Since a logarithm of a nonpositive number cannot be taken,
Therefore, the vertical asymptote must be the vertical line of the equation
Each of Statement 1 and Statement 2 gives us only one of
making the vertical asymptote
← Didn't Know|Knew It →