GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Graphing

Define a function as follows:

$g(x) =2 \log_{4}(x-4)+ 3$

Give the -intercept of the graph of .

Possible Answers:

(12,0)

The graph of has no -intercept.

(-4, 0)

$\left ( 3\frac{7}{8}, 0 \right )$

$\left ( 4\frac{1}{8}, 0 \right )$

Correct answer:

$\left ( 4\frac{1}{8}, 0 \right )$

Explanation:

Set g(x) =0 and evaluate to find the -coordinate of the -intercept.

$g(x) =2 \log_{4}(x-4)+ 3 = 0$

$2 \log_{4}(a-4) = -3$

$\log_{4}(a-4) = -\frac{3}{2}$

This can be rewritten in exponential form:

$a - 4 = 4 ^{-\frac{3}{2}}= \frac{1}{4 ^{ \frac{3}{2}}} = \frac{1}{(\sqrt{4 })^{ 3}} = \frac{1}{2^{ 3}} = \frac{1}{8}$

$a = 4\frac{1}{8}$

The -intercept of the graph of is $\left ( 4\frac{1}{8}, 0 \right )$ .

Report an Error

Example Question #2 : Graphing

Define a function as follows:

$g(x) = \log_{9}(x-4)+ 2$

Give the $y\:$ -intercept of the graph of .

Possible Answers:

$\left ( 0, 3\frac{80}{81}\right )$

(0, 2)

$\left ( 0, 4\frac{1}{81}\right )$

The graph of has no $y\:$ -intercept.

(0,4)

Correct answer:

The graph of has no $y\:$ -intercept.

Explanation:

The $y\:$ -coordinate of the $y\:$ -intercept is g(0) :

$g(x) = \log_{9}(x-4)+ 2$

$g(0) = \log_{9}(0-4)+ 2$

$= \log_{9}( -4)+ 2$

However, the logarithm of a negative number is an undefined expression, so g(0) is an undefined quantity, and the graph of has no $y\:$ -intercept.

Report an Error

Example Question #71 : Graphing

Define a function as follows:

$g(x) =-2 \log_{8}(x+2)+ 3$

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

Possible Answers:

$x = \frac{2}{3}$

x = -2

x = 2

x = -3

x = 3

Correct answer:

x = -2

Explanation:

Only positive numbers have logarithms, so

x+2 > 0

x> -2

The graph never crosses the vertical line of the equation x = -2 , so this is the vertical asymptote.

Report an Error

Example Question #72 : Graphing

Define a function as follows:

$g(x) =2 \ln (x-4)+ 3$

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

Possible Answers:

The graph of has no vertical asymptote.

x = 3

$x = -\frac{3}{2}$

x = 4

x = 2

Correct answer:

x = 4

Explanation:

Only positive numbers have logarithms, so

x- 4 > 0

x> 4

The graph never crosses the vertical line of the equation x = 4 , so this is the vertical asymptote.

Report an Error

Example Question #73 : Graphing

Define a function as follows:

$g(x) =-2 \log_{8}(x+2)+ 3$

Give the $y\:$ -intercept of the graph of .

Possible Answers:

$\left ( 2\frac{1}{3},0 \right )$

The graph of has no $y\:$ -intercept.

(-3,0)

$\left ( 3\frac{2}{3},0 \right )$

(9,0)

Correct answer:

$\left ( 2\frac{1}{3},0 \right )$

Explanation:

The $y\:$ -coordinate of the $y\:$ -intercept is g(0) :

$g(x) =-2 \log_{8}(x+2)+ 3$

$g(0) =-2 \log_{8}(0 +2)+ 3$

$g(0) =-2 \log_{8}2+ 3$

Since 2 is the cube root of 8, $8^{\frac{1}{3}}= 2$ , and $\log_{8}2 = \frac{1}{3}$ . Therefore,

$g(0) =-2 \cdot \frac{1}{3}+ 3 = -\frac{2}{3}+ 3 = 2\frac{1}{3}$ .

The $y\:$ -intercept is $\left ( 2\frac{1}{3},0 \right )$ .

Report an Error

Example Question #1051 : Gmat Quantitative Reasoning

Define functions and as follows:

$f(x) = \log \frac{1}{x+3}$

$g(x) = \log (4x+12)$

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

Possible Answers:

$-\log 2$

The graphs of and do not intersect.

$\log 2$

Correct answer:

$\log 2$

Explanation:

Since $- \log N = \log \left ( \frac{1}{N} \right )$ , the definition of can be rewritten as follows:

$f(x) = - \log (x+3)$

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

g(x)= f(x)

$\log (4x+12)= \log \frac{1}{x+3}$

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

$(4x+12)= \frac{1}{x+3}$

$(4x+12)(x+3)= \frac{1}{x+3} \cdot (x+3)$

$4x^{3}+24x+36 =1$

$4x^{3}+24x+35 =0$

We can solve using the method, finding two integers whose sum is 24 and whose product is $4 \cdot 35 = 140$ - these integers are 10 and 14, so we split the niddle term, group, and factor:

$(4x^{3}+10x)+(14x+35 )=0$

2x (2x+5)+7(2x+5 )=0

(2x +7)(2x+5 )=0

2x+7 = 0

2x= -7

$x = -\frac{7}{2} = -3\frac{1}{2}$

2x+5= 0

2x= -5

$x = -\frac{5}{2} = -2\frac{1}{2}$

This gives us two possible -coordinates. However, since

$g\left ( -3\frac{1}{2} \right ) = \log \left [4\left ( -3\frac{1}{2} \right ) +12 \right ] = \log (-14+12) = \log (-2)$ ,

an undefined quantity - negative numbers not having logarithms -

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

$f\left ( -2\frac{1}{2} \right ) = \log \frac{1}{ -2\frac{1}{2} +3} = \log \frac{1}{\frac{1}{2}} = \log 2$

and

$g\left ( -2\frac{1}{2} \right ) = \log \left [4\left ( -2\frac{1}{2} \right ) +12 \right ] = \log (-10+12) = \log 2$

$-2 \frac{1}{2}$ is the correct -value, and $\log 2$ is the correct -value.

Report an Error

Example Question #3 : Graphing

Let (x,y) be the point of intersection of the graphs of these two equations:

$2 \log_{2} x-3 \log_{3} y = 13$

$5 \log_{2} x+2 \log_{3} y = 4$

Evaluate $y\:$ .

Possible Answers:

y = -27

$y = \frac{1}{9}$

$y = \frac{1}{27}$

y = 9

Correct answer:

$y = \frac{1}{27}$

Explanation:

Substitute and for $\log_{2} x$ and $\log_{3} y$ , respectively, and solve the resulting system of linear equations:

2 u-3 v = 13

5 u+2 v = 4

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

4 u-6 v = 26

$\underline{15 u+6 v = 12}$

19u =38

$u = 38 \div 19 = 2$

Now back-solve:

5 (2)+2 v = 4

10+2 v = 4

2v=-6

v= -3

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

u = 2

$\log_{2} x = 2$

$x = 2^{2} = 4$ .

v= -3

$\log_{3} y = -3$

$y = 3^{-3} = \frac{1}{3^{3}} = \frac{1}{27}$

$\left (4, \frac{1}{27} \right )$ is the solution, and $y = \frac{1}{27}$ , the correct choice.

Report an Error

Example Question #11 : Graphing

Let (x,y) be the point of intersection of the graphs of these two equations:

$2 \log x+3 \log y = 12$

$5 \log x- 2 \log y = 11$

Evaluate .

Possible Answers:

x = 100

The system has no solution.

x = 1,000

$x =\frac{1}{ 1 00}$

$x =\frac{1}{ 1,000}$

Correct answer:

x = 1,000

Explanation:

Substitute and for $\log x$ and $\log y$ , respectively, and solve the resulting system of linear equations:

2 u+3 v = 12

5 u- 2 v = 11

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

4 u+6 v = 24

$\underline{15 u- 6 v = 33}$

19u =57

$u = 57 \div 19 = 3$

Back-solve:

2 (3)+3 v = 12

6+3 v = 12

3v = 6

v = 2

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

u = 3

$\log x = 3$

$x=10^{3} = 1,000$

and

v=2

$\log y = 2$

$y=10^{2} = 100$

We check this solution in both equations:

$2 \log x+3 \log y = 12$

$2 \log 1,000+3 \log 100 = 12$

2 (3)+3 (2)= 12

6+6 = 12 - true.

$5 \log x- 2 \log y = 11$

$5 \log 1,000- 2 \log 100 = 11$

5 (3) -2 (2) = 11

15-4 = 11 - true.

(1000, 100) is the solution, and x=1,000 , the correct choice.

Report an Error

Example Question #12 : Graphing

The graph of function has vertical asymptote x = 5 . Which of the following could give a definition of ?

Possible Answers:

$f(x) =7\log (x-3)+5$

$f(x) =3\log (x-7)-5$

$f(x) = 3 \log (x-5)+7$

$f(x) = -5 \log (x-7)-3$

$f(x) = 5 \log (x-3)-7$

Correct answer:

$f(x) = 3 \log (x-5)+7$

Explanation:

Given the function $f(x) = A \log (x-B)+C$ , 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 x-B >0 , or, equivalently, x>B and that the graph of will never cross the vertical line x=B . That makes x=B the vertical asymptote, so it follows that the graph with vertical asymptote x = 5 will have in the position. The only choice that meets this criterion is

$f(x) = 3 \log (x-5)+7$

Report an Error

Example Question #13 : Graphing

The graph of a function has -intercept (64,0) . Which of the following could be the definition of ?

Possible Answers:

$f(x)= \log_{16}x- \frac{3}{2}$

$f(x)= \log_{2}x-6$

$f(x)= \log_{4}x-3$

All of the other choices are correct.

$f(x)= \log_{8}x-2$

Correct answer:

All of the other choices are correct.

Explanation:

All of the functions are of the form $f(x) = \log_{b}x - C$ . To find the -intercept of such a function, we can set f(x) = 0 and solve for :

$\log_{b}x - C = 0$

$\log_{b}x =C$

$b^{C}= x$

Since we are looking for a function whose graph has -intercept (64,0) , the equation here becomes $b^{C}= 64$ , and we can examine each of the functions by finding the value of $b^{C}$ .

$f(x)= \log_{8}x-2$ :

b= 8, c= 2

$b^{C}= 8^{2} = 64$

$f(x)= \log_{2}x-6$ :

b = 2,c=6

$b^{C}= 2^{6} = 64$

$f(x)= \log_{4}x-3$ :

b=4,c=3

$b^{C}=4^{3} = 64$

$f(x)= \log_{16}x- \frac{3}{2}$ :

$b=16,c=\frac{3}{2}$

$b^{C}= 16^{ \frac{3}{2}} = \left ( \sqrt{16} \right )^{3}= 4^{3}= 64$

All four choices fit the criterion.

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Denver, SSAT Tutors in Miami, LSAT Tutors in Philadelphia, ISEE Tutors in Chicago, ACT Tutors in Chicago, Biology Tutors in Denver, Spanish Tutors in Houston, Math Tutors in Dallas Fort Worth, Algebra Tutors in Atlanta, LSAT Tutors in Atlanta

Popular Courses & Classes

Spanish Courses & Classes in Philadelphia, MCAT Courses & Classes in Philadelphia, MCAT Courses & Classes in Boston, SAT Courses & Classes in Phoenix, LSAT Courses & Classes in Phoenix, ISEE Courses & Classes in Chicago, ACT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Washington DC, ISEE Courses & Classes in Los Angeles

Popular Test Prep

GRE Test Prep in Miami, LSAT Test Prep in Boston, ACT Test Prep in Dallas Fort Worth, GRE Test Prep in Washington DC, ACT Test Prep in Atlanta, LSAT Test Prep in Washington DC, GMAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in San Diego, GMAT Test Prep in Chicago, SAT Test Prep in Washington DC

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2 : Graphing

Example Question #2 : Graphing

Example Question #71 : Graphing

Example Question #72 : Graphing

Example Question #73 : Graphing

Example Question #1051 : Gmat Quantitative Reasoning

Example Question #3 : Graphing

Example Question #11 : Graphing

Example Question #12 : Graphing

Example Question #13 : Graphing

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details