Last visit was: 19 Jul 2025, 20:41 It is currently 19 Jul 2025, 20:41
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
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
Back to Forum
Create Topic
705-805 Level|   Algebra|   Exponents|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Jul 2025
Posts: 102,627
Own Kudos:
742,808
 [72]
Given Kudos: 98,235
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,627
Kudos: 742,808
 [72]
If 2^x = 3, then which of the following must be true? 06 May 2020, 10:31
1
Kudos
Add Kudos
71
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

75% (hard)

Question Stats:

53% (02:14) correct 47% (01:58) wrong based on 475 sessions

History

Date
Time
Result
Not Attempted Yet
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


M37-42

Experience a GMAT Club Test Questions
Yes, you've landed on a GMAT Club Tests question
Craving more? Unlock our full suite of GMAT Club Tests here
Want to experience more? Get a taste of our tests with our free trial today
Rise to the challenge with GMAT Club Tests. Happy practicing!
GMAT Club Tests Check Our Products Try Free Tests
ShowHide Answer
Official Answer
B
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Jul 2025
Posts: 102,627
Own Kudos:
742,808
 [6]
Given Kudos: 98,235
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,627
Kudos: 742,808
 [6]
If 2^x = 3, then which of the following must be true? 24 Nov 2021, 07:34
2
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
Official Solution:

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


Square: \(2^{2x} = 3^2=9\);

\((2^{2x} = 9) > 2^3\), so \(2x > 3\);

\(x>1 \frac{1}{2}\)

Cube: \(2^{3x} = 3^3=27\);

\((2^{3x} = 27) < 2^5\), so \(3x < 5\);

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

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


Answer: B
General Discussion
User avatar
TestPrepUnlimited
User avatar
Joined: 17 Sep 2014
Last visit: 30 Jun 2022
Posts: 1,226
Own Kudos:
1,067
 [16]
Given Kudos: 6
Location: United States
GMAT 1: 780 Q51 V45
GRE 1: Q170 V167
Expert
Expert reply
GMAT 1: 780 Q51 V45
GRE 1: Q170 V167
Posts: 1,226
Kudos: 1,067
 [16]
If 2^x = 3, then which of the following must be true? 06 May 2020, 10:42
4
Kudos
Add Kudos
12
Bookmarks
Bookmark this Post
Bunuel
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
User avatar
yashikaaggarwal
User avatar
Senior Moderator - Masters Forum
Joined: 19 Jan 2020
Last visit: 17 Jul 2025
Posts: 3,095
Own Kudos:
3,046
Given Kudos: 1,510
Location: India
Schools: Cranfield (A) Warwick MiM "23 (M)
GPA: 4
WE:Analyst (Internet and New Media)
Schools: Cranfield (A) Warwick MiM "23 (M)
Posts: 3,095
Kudos: 3,046
If 2^x = 3, then which of the following must be true? 06 May 2020, 10:45
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 28 Jun 2025
Posts: 1,853
Own Kudos:
7,843
 [1]
Given Kudos: 707
Location: India
Posts: 1,853
Kudos: 7,843
 [1]
If 2^x = 3, then which of the following must be true? 06 May 2020, 19:06
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
\(x = log_2 3\)

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

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

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

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
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
AnirudhaS
User avatar
LBS Moderator
Joined: 30 Oct 2019
Last visit: 25 Jun 2024
Posts: 812
Own Kudos:
850
Given Kudos: 1,575
Posts: 812
Kudos: 850
If 2^x = 3, then which of the following must be true? 07 May 2020, 12:47
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 28 Jun 2025
Posts: 1,853
Own Kudos:
7,843
 [1]
Given Kudos: 707
Location: India
Posts: 1,853
Kudos: 7,843
 [1]
If 2^x = 3, then which of the following must be true? 07 May 2020, 19:23
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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
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
satya2029
User avatar
Joined: 10 Dec 2017
Last visit: 19 Jul 2025
Posts: 231
Own Kudos:
242
Given Kudos: 137
Location: India
Posts: 231
Kudos: 242
If 2^x = 3, then which of the following must be true? 08 May 2020, 04:10
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
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
Bambi2021
User avatar
Joined: 13 Mar 2021
Last visit: 23 Dec 2021
Posts: 321
Own Kudos:
131
Given Kudos: 226
Posts: 321
Kudos: 131
If 2^x = 3, then which of the following must be true? 10 Jul 2021, 15:05
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Must admit I failed this question. But I found some valuable approximations of x. The fractions represent the closest distances between a power of 2 and a power of 3.

https://oeis.org/A254351/internal

"log(3)/log(2) = 1.5849625... is an irrational number. The fractions (2/1, 3/2, 5/3, 8/5, 11/7, 19/12, 46/29, 65/41, 84/53, 317/200, 401/253, 485/306, 569/359, 1054/665, ...) are a sequence of approximations to log(3)/log(2), where each is an improvement on its predecessors."

This means that 2^(19/12) is closer to 3 than 2^(11/7).

"Seven/eleven" seems easy to remember as a decent approximation. =)

Posted from my mobile device
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 37,453
Own Kudos:
1,013
Posts: 37,453
Kudos: 1,013
If 2^x = 3, then which of the following must be true? 24 Jun 2025, 00:22
Kudos
Add Kudos
Bookmarks
Bookmark this Post
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.
Moderators:
Bunuel
Math Expert
102627 posts
Krunaal
PS Forum Moderator
698 posts