If the expression x^x^x^(...), where the given expression expression

Question

If x > 0 and the expression  x^{x^{x^{...}}} , where the given expression extends to an infinite number of exponents, equals to 2, then what is the value of x?

A. 1/2

B.  \sqrt {2}

C.  \sqrt{2}

D.  \sqrt{3}

E. 2

Official Solution is    HERE .

Bunuel · Accepted Answer

Official Solution:

If  x>0  and the expression  x^{x^{x^{...}}} , where the given expression extends to an infinite number of exponents, equals to 2, then what is the value of  x ?

A.  \frac{1}{2} 
B.  \sqrt {2} 
C.  \sqrt{2} 
D.  \sqrt{3} 
E.  2

Re-write as:  x^{(x^{x^{...}})} . Since the expression extends to an infinite number of exponents, then the expression in brackets would also equal to 2. Thus we can replace the expression in brackets with 2 and rewrite the given expression as  x^2=2 . So,  x=\sqrt{2}

Answer: C

vitaliyGMAT · Answer

Hi, nice question.

x^{x^{x}^{...}} = 2

lnx^{x^{x}^{...}} = ln2

x^{x^{x}^{...}}*lnx = ln2

2*lnx = ln2

lnx = \frac{1}{2} * ln2

lnx = ln \sqrt{2}

x = \sqrt{2}

Answer C

GMATAcademy · Answer

Because the pattern continues infinitely, the exponent to which the first x is raised is identical to the overall expression.

We can use this information to write an equation:

(overall expression) = x^(overall expression) = 2

or just:

x^(overall expression) = 2

Since we know the overall expression equals 2, we can plug in:

x^(2) = 2

C is the only answer that satisfies the equation: when we square root 2, we get 2.

falewman · Answer

Soln:

x^{x^{x}^{...}}  = 2

We are given that X raised to  x^{x^{x}^{...}}  = 2;

Since the powers of X extends to infinity, we can retain the base 'X' and replace the whole power by 2. This reduces the equation to a simple expression of:

x^{2}  = 2

Hence X = \sqrt{2}

maxschmid · Answer

Can somebody please explain why the answer is E and not C?

chetan2u · Answer

Hi..

x^{x^{x}^{...}}=2 ....
Since it is to infinite series,  x^{x^{x}^{...}}=x^2=2 ,
Or  x=\sqrt{2} 
C..

Bunuel , pl check the OA, must be a typo error

Bunuel · Answer

Yes, it's C. Edited the OA.

Bunuel · Answer

Similar questions to practice: https://gmatclub.com/forum/tough-and-tri ... l#p1029228 https://gmatclub.com/forum/find-the-valu ... 38049.html https://gmatclub.com/forum/find-the-valu ... 75403.html https://gmatclub.com/forum/if-the-expre ... 98647.html https://gmatclub.com/forum/m26-184445.html https://gmatclub.com/forum/if-the-expre ... 32547.html

shashankism · Answer

Let y =  x^{x^{x}^{...}}  = 2
So, y = x^y = 2
Taking log on both sides
 log y = y log x 
 log 2 = 2 log x 
 log x = 1/2 log 2 = log (\sqrt{2}) 
 x = \sqrt{2}

sasidharrs · Answer

does GMAT tests natural logorithms?

Bunuel · Answer

You don't need this staff at all. If you check  official solution above  you'll see that this question can be solved without logarithms.

shashankism · Answer

Bunuel  gave very good solution and now I also feel that it can be solved without using natural log. 
I don't know why I used log , probably because we always use log in such types of problem but it can be solved easily by using the method explained by bunuel.

It goes like this :
 y = x^x^x^x^{....} = 2 
 So, x^2 = 2 
 x = \sqrt{2}

Madhavi1990 · Answer

Interesting one - plays on wording and what we make of it.
If x > 0 and the expression x^{x^{x}^{...}}, where the given expression extends to an infinite number of exponents, equals to 2, then what is the value of x?

Basically the power of x ^ x^x...so the exponent = 2. It also says x =2. So by plugging base and exponent we have x^2 = 2 (cause x =2; stated earlier in question). Which is root 2.

A. 1/2

B. 2√424

C. 2√2

D. 3√3

E.

jasonfodor · Answer

Hi, can you explain why:

Since the expression extends to an infinite number of exponents, then the expression in brackets would also equal to 2.

How did you come up with this? Why can't it equal say 3 or 0 or 4923.3 or any other number.

Bunuel · Answer

We know that  x^{x^{x^{...}}}=2 , where x's are infinite. Infinite - 1 is also infinite, so  x^2=2 .

wardalaknaoui1 · Answer

the powers of X extends to infinity, we can retain the base 'X' and replace the whole power by 2.
I am sorry how did you come up with that rule , I read all the comments and all of you say the same thing, but i am not able to get it yet.

svsivakrishna · Answer

HI

You can try to put it this way inorder to understand:

Given expression: x^x^x^(...) =2 and the powers extend to infinite x's

So, reducing one x out of infinity also gives infinite x (meaning: Doesnt make a difference.)

Therefore x^2=2 and x= sqrt (2).

Hope this would have given a slightly different understanding.

If the expression x^x^x^(...), where the given expression expression

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If the expression x^x^x^(...), where the given expression expression

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards