GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
close
It is currently 14 Jun 2021, 13:43
Register
GMAT Club Tests
Decision Tracker
My Rewards
New posts
New comers' posts

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Go to My Error Log Learn more
Close

Request Expert Reply

Confirm Cancel
MBA Spotlight – Top 20 Virtual MBA Fair
June 14 - 30, 2021
schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos schools-logos
MBA Spotlight is a Virtual MBA Fair bringing together MBA applicants and Admissions Teams from the leading Business Schools. Join GMAT Club in a Live Q&A with Admissions Directors of the World's Top MBA programs and Current Students. Every day at 10 AM PDT, join for live Zoom Breakouts and profile evaluations by Experts to help with your application strategy.
Register Schedule
If 2^x = 3, then which of the following must be true?
8 posts Go to First Unread Post
Print view
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
SORT BY:
Date
Kudos
Tags:
Find Similar Topics 

Show Tags
Hide Tags

User avatar
V
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 77064
Premium Member CAT Tests Top Member of the Month
If 2^x = 3, then which of the following must be true? [#permalink] New post  06 May 2020, 09:31
24
Bookmarks
Expert Reply
Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
00:00
A
B
C
D
E

Difficulty:

75% (hard)

Question Stats:

54% (02:14) correct 46% (01:53) wrong based on 134 sessions

Hide Show timer Statistics

If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

Are You Up For the Challenge: 700 Level Questions
ShowHide Answer
Official Answer
Official Answer and Stats are available only to registered users. Register/Login.

_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
Attempt only this question Same topic and source Same topic and difficulty Same source and difficulty Random Chronological Reverse chronological Learn more
Most Helpful Expert Reply
User avatar
G
TestPrepUnlimited
GMAT Tutor
Joined: 16 Sep 2014
Posts: 1112
Location: United States
GMAT 1: 780 Q51 V45
GRE 1: Q170 V167
If 2^x = 3, then which of the following must be true? [#permalink] New post  06 May 2020, 09:42
4
Kudos
5
Bookmarks
Expert Reply
Bunuel wrote:
If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

Are You Up For the Challenge: 700 Level Questions


We can do some manipulations to help us determine how big \(x\) is in \(2^x = 3\).

If we square both sides we can get \(2^{2x} = 3^2 = 9\). 9 is a little bit bigger than \(2^3\). So we have \(2^{2x} > 2^3\). It follows that \(2x > 3\) and \(x > \frac{3}{2}\).

We could choose B here since we know x should be just a little bit bigger than \(\frac{3}{2}\), but if we want to be safe we can try to find an upper bound for x.

Let's try using the fact that \(2^{3x} = 3^3 = 27\). We want an upper bound this time so we have \(2^{3x} < 32 = 2^5\) and \(3x < 5\), which is \(x < \frac{5}{3}\). Hence we can comfortably choose B.

Ans: B
_________________
Source: We are an NYC based, in-person and online GMAT tutoring and prep company. We are the only GMAT provider in the world to guarantee specific GMAT scores with our flat-fee tutoring packages, or to publish student score increase rates. Our typical new-to-GMAT student score increase rate is 3-9 points per tutoring hour, the fastest in the world. Feel free to reach out!
Signature Read More
General Discussion
User avatar
V
yashikaaggarwal
Senior Moderator - Masters Forum
Joined: 18 Jan 2020
Posts: 2539
Location: India
Schools: HEC MiM "23, ESSEC MiM "23
GPA: 4
Premium Member CAT Tests
Re: If 2^x = 3, then which of the following must be true? [#permalink] New post  06 May 2020, 09:45
Answer is supposedly B.
A. 2^4/3= 2.51 < x < 2^3/2= 2.82 (less than 2^x=3)

B. 2^3/2 =2.82 < x < 2^5/3= 3.17 (3 lies in between)

C. 2^5/3= 3.17 < x < 2^7/4= 3.36 (More than 2^x=3)

D. 2^7/4= 3.36 < x < 2^11/6= 3.56 (More than 2^x=3)

E. 2^11/6= 3.56 < x < 2^2= 4 (More than 2^x=3)

B is the answer

Posted from my mobile device
User avatar
V
nick1816
DS Forum Moderator
Joined: 19 Oct 2018
Posts: 2290
Location: India
Re: If 2^x = 3, then which of the following must be true? [#permalink] New post  06 May 2020, 18:06
1
Kudos
\(x = log_2 3\)

\(x= \frac{log_{10}3}{log_{10}2}\)

\(x= \frac{0.47}{0.3} \)

\(1.6>\frac{0.47}{0.3}>1.5\)

OR

\(2^{1.5} = 2^{(1+0.5)} = 2*2^{0.5} = 2*1.414 = 2.82\)

\(2^{1+\frac{2}{3}} = 2*2^{\frac{2}{3}} = 2*4^{\frac{1}{3}} = 2*[\frac{500}{125}]^{\frac{1}{3}} \) ~ \(2*[\frac{512}{125}]^{\frac{1}{3}}\)= \(2*(\frac{8}{5}) = 3.2\)



Or

\((2^3)*(\frac{9}{8}) = 3^2\)

\(2^3*1.125 = 3^2\)

We know that 2^0= 1 and 2^1 = 2

1.125 must be in between \(2^{0.1}\) and \(2^{0.2}\)

\(2^3*2^{0.1} < 2^{2x} < 2^3*2^{0.2}\)

\(2^{3.1} < 2^{2x} < 2^{3.2}\)

\(2^{1.55}< 2^x< 2^{1.6}\)

B






Bunuel wrote:
If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

Are You Up For the Challenge: 700 Level Questions
User avatar
V
AnirudhaS
LBS Moderator
Joined: 30 Oct 2019
Posts: 798
Location: United Kingdom
Concentration: General Management, Technology
Schools: INSEAD Jan'21 Intake, Cambridge '22, LBS '23
GPA: 4
Re: If 2^x = 3, then which of the following must be true? [#permalink] New post  07 May 2020, 11:47
I solved this by "feeling". Let me explain. But I need experts to critic me on this method. nick1816 let me know your thoughts please?

So we know
2^1.1
2^1.2
2^1.3
.....
2^1.9
2^2

is going to be an exponential curve (NOT linear).

And if it was linear then 2^1.5 would be exactly 3, but its not.
So 2^1.5 < 3
But the rate of growth of this exponential curve is only slight (not as much as say 2^2, 2^3, 2^4...)
So 2^1.5 should be quite close to 3

I mean then I chose the option B which is closest. However, I do understand that this is not full proof (as we do not know how close 2^1.5 is to 3 unless we work it out)
User avatar
V
nick1816
DS Forum Moderator
Joined: 19 Oct 2018
Posts: 2290
Location: India
If 2^x = 3, then which of the following must be true? [#permalink] New post  07 May 2020, 18:23
1
Kudos
You are on point bhai. Since the curve is exponential, 2^1.5 < 3.

Now you could find the approximate value of \(2^{(1\frac{2}{3})}\) just to make sure that you're marking the correct option. I've already mentioned one way to approximate \(2^{(1\frac{2}{3})}\)in my solution.

Another way is-

\(3375 < 4000 < 4096\)

\(15^3<4000<16^3\)

\(1.5^3 < 4 < 1.6^3\)

\(1.5 < 4^{(\frac{1}{3})} < 1.6\)

\(1.5 < 2^{(\frac{2}{3})} < 1.6\)

\(1.5*2 < 2*2^{(\frac{2}{3})} < 1.6*2\)

\(3 < 2^1*2^{(\frac{2}{3})} < 3.2\)

\(3 < 2^{(1+\frac{2}{3})} < 3.2\)



AnirudhaS wrote:
I solved this by "feeling". Let me explain. But I need experts to critic me on this method. nick1816 let me know your thoughts please?

So we know
2^1.1
2^1.2
2^1.3
.....
2^1.9
2^2

is going to be an exponential curve (NOT linear).

And if it was linear then 2^1.5 would be exactly 3, but its not.
So 2^1.5 < 3
But the rate of growth of this exponential curve is only slight (not as much as say 2^2, 2^3, 2^4...)
So 2^1.5 should be quite close to 3

I mean then I chose the option B which is closest. However, I do understand that this is not full proof (as we do not know how close 2^1.5 is to 3 unless we work it out)
avatar
P
satya2029
Senior Manager
Senior Manager
Joined: 10 Dec 2017
Posts: 279
Location: India
If 2^x = 3, then which of the following must be true? [#permalink] New post  08 May 2020, 03:10
Bunuel wrote:
If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

Are You Up For the Challenge: 700 Level Questions


2^x=3
2^x-2=1
2^(x-1)=3/2=1.5
\(\sqrt{2}=1.414\)
0.5<x<0.6
B:)
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 19424
Re: If 2^x = 3, then which of the following must be true? [#permalink] New post  30 May 2021, 08:49
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources
Signature Read More
GMAT Club Bot
gmatclubot
Re: If 2^x = 3, then which of the following must be true? [#permalink]
30 May 2021, 08:49
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
Moderators:
chetan2u
Math Expert
9996 posts
Bunuel
Math Expert
77064 posts
yashikaaggarwal
Senior Moderator - Masters Forum
2539 posts
generis
Senior SC Moderator
4798 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Main navigation

GMAT Resources

Partners

MBA Resources

www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions