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If x is an integer and 4^x < 100, what is x?

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If x is an integer and 4^x < 100, what is x?  [#permalink]

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If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100
(2) 4^(x + 1) + 4^x > 100

Originally posted by rite2deepti on 16 Dec 2010, 04:37.
Last edited by Bunuel on 24 May 2017, 21:26, edited 1 time in total.
Edited the OA.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 06:27
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4
rite2deepti wrote:
If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100
(2) 4^(x + 1) + 4^x > 100
I dont have the OA for this question..I need help ...Thanks



Given: x is an integer and \(4^x < 100\)
So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions.
\(4^{x - 1}*(4^2 - 1) > 100\)
\(4^{x - 1}*15 > 100\)
For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\).
\(4^x*(4 + 1) > 100\)
\(4^x*(5) > 100\)
For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 05:08
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rite2deepti wrote:
If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100
(2) 4^(x + 1) + 4^x > 100
I dont have the OA for this question..I need help ...Thanks



answer is D as both statements are sufficient to answer the question.

For 4^x < 100 to hold true X has to be less than Four, X< 4.

(1) 4^(x + 1) – 4^(x – 1) > 100 or it can be simplified to 15 * 4^(x-1) > 100

choose a numbers and work out which number for X will satisfy this statement.
(i) choose number 1 answer will be <100
(ii) choose number 2 and answer will be <100
(iii) choose number 3 and answer will be >100
Therefore the the value of X>=3
But for 4^x < 100 to hold true X has to be 3. SUFFICIENT

(2) 4^(x + 1) + 4^x > 100 which can be simplified to 5* 4^x > 100 or to 4^x > 20
choose a numbers and work out which number for X will satisfy this statement.
i) choose number 1 answer will be <100
(ii) choose number 2 and answer will be < 100
(iii) choose number 3 and answer will be >100
Therefore the the value of X>=3
But for 4^x < 100 to hold true X has to be 3. SUFFICIENT
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 05:16
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1
If x is an integer and 4^x < 1000, what is x?

Given: \(x=integer\) and \(4^x<1,000\) --> \(2^{2x}<1,000\) --> \(2x<10\) (as \(2^{10}=1024\)) --> \(x<5\), as \(x\) is an integer then it's cam be 4, 3, 2, 1, 0, -1, ...

(1) 4^(x + 1) – 4^(x – 1) > 100 --> factor out \(4^x\) ---> \(4^x(4-\frac{1}{4})>100\) --> \(4^x>\frac{400}{15}\approx{27}\) --> \(x>2\), so \(x\) is either 3 or 4. Not sufficient.
(2) 4^(x + 1) + 4^x > 100 --> --> factor out \(4^x\) ---> \(4^x(4+1)>100\) --> \(4^x>20\) --> \(x>2\), so \(x\) is either 3 or 4. Not sufficient.

(1)+(2) The same 2 values for \(x\): 3 and 4. Not sufficient.

Answer: E.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 05:18
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Thanks Bunuel....You saved me from getting confused ..the answer given by tirupatibalaji was wrong....Thanks a lot ..
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 05:25
2
Bunnel
But the original question is 4^x < 100, what is x?
But not
4^x < 1000, what is x?

Or did i miss something?
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 05:40
2
1
tirupatibalaji wrote:
Bunnel
But the original question is 4^x < 100, what is x?
But not
4^x < 1000, what is x?

Or did i miss something?


Yes, I solved for 4^x < 1,000. Maybe because in this case question becomes a little bit harder.

As actually it's 4^x < 100 then from stem we'll have x<4, so x can be 3, 2, 1, ...

And as both statements give x>2 then x can only be 3. So both statements alone are sufficient.

Answer: D.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 16 Dec 2010, 06:51
Bunuel wrote:
tirupatibalaji wrote:
Bunnel
But the original question is 4^x < 100, what is x?
But not
4^x < 1000, what is x?

Or did i miss something?


Yes, I solved for 4^x < 1,000. Maybe because in this case question becomes a little bit harder.

As actually it's 4^x < 100 then from stem we'll have x<4, so x can be 3, 2, 1, ...

And as both statements give x>2 then x can only be 3. So both statements alone are sufficient.

Answer: D.


Thanks Bunuel.

I thought I was TOTALLY wrong, but this edit makes me feel better!
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 01 Nov 2015, 13:12
2
use common logarithm is fast, simple and very direct:

4^x<100

so 2xlg2<2, lg2=0.301, so x<=3

what is x?

(1) 4^x+1-4^x-1>100

so (15/4)*4^x>100

so 2xlg2>3lg2+1-lg3, lg3=0.477, so x>=3

sufficient

(2) 4^x+1+4^x>100

so 5*4^x>100

4^x>20

so 2xlg2>lg2+1

x>=3

sufficient

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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 03 Nov 2015, 20:55
VeritasPrepKarishma wrote:
rite2deepti wrote:
If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100
(2) 4^(x + 1) + 4^x > 100
I dont have the OA for this question..I need help ...Thanks



Given: x is an integer and \(4^x < 100\)
So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions.
\(4^{x - 1}*(4^2 - 1) > 100\)
\(4^{x - 1}*15 > 100\)
For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\).
\(4^x*(4 + 1) > 100\)
\(4^x*(5) > 100\)
For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).


Can someone explain simplifying the equation by factoring out 4^x and so on. I can't seem to wrap my head around it right now.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 05 Nov 2015, 00:10
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garey8831 wrote:

Can someone explain simplifying the equation by factoring out 4^x and so on. I can't seem to wrap my head around it right now.


When you have addition/subtraction between terms with exponents, all you can do is take common.

Can you simplify \(4^5 + 4^6\)?
Note that \(4^5 = 4*4*4*4*4\)
and \(4^6 = 4*4*4*4*4*4\)

\((4*4*4*4*4) + (4*4*4*4*4*4)\)

You can take five 4s common. You will get

\(4*4*4*4*4 ( 1 + 4)\)
\(= 4^5 * 5\)

So here, it is the same concept.

\(4^x + 4^{x+1}\)
You take \(4^x\) common.
\(4^x ( 1 + 4)\)
\(4^x * 5\)

and

\(4^{x+1} - 4^{x-1}\)
You take \(4^{x-1}\) common because the exponent (x-1) is smaller.
\(4^{x-1} ( 4*4 - 1)\) (because \(4^{x+1}\) has two extra 4s so they are left behind}
\(4^{x-1} * 15\)
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 02 Sep 2016, 12:39
VeritasPrepKarishma wrote:
rite2deepti wrote:
If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100
(2) 4^(x + 1) + 4^x > 100
I dont have the OA for this question..I need help ...Thanks



Given: x is an integer and \(4^x < 100\)
So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions.
\(4^{x - 1}*(4^2 - 1) > 100\)
\(4^{x - 1}*15 > 100\)
For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\).
\(4^x*(4 + 1) > 100\)
\(4^x*(5) > 100\)
For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).


Since the questions just mentions that x is an integer, then why can't x be smaller than -3 such as -4, -5 and so on? Because \(4^{-x} = \frac{1}{4^x}\). Am i missing something here?
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 03 Sep 2016, 02:29
1
sam2016 wrote:
VeritasPrepKarishma wrote:
rite2deepti wrote:
If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) – 4^(x – 1) > 100
(2) 4^(x + 1) + 4^x > 100
I dont have the OA for this question..I need help ...Thanks



Given: x is an integer and \(4^x < 100\)
So x could be 3/2/1/0/-1/-2/-3... For each of these values, \(4^x < 100\)

Question: what is x?

Stmnt 1: \(4^{x+1} - 4^{x - 1} > 100\)

From above, factor out \(4^{x - 1}\). Try and factor out the smallest index to avoid dealing with fractions.
\(4^{x - 1}*(4^2 - 1) > 100\)
\(4^{x - 1}*15 > 100\)
For this product to be greater than 100, \(4^{x - 1}\) should be at least \(4^2\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Stmnt 2: \(4^{x+1} + 4^x > 100\)

From above, factor out \(4^x\).
\(4^x*(4 + 1) > 100\)
\(4^x*(5) > 100\)
For this product to be greater than 100, \(4^x\) should be at least \(4^3\) or x must be 3 or greater.
Since x is one of 3/2/1/0/-1/....., x must be 3. Sufficient.

Answer (D).


Since the questions just mentions that x is an integer, then why can't x be smaller than -3 such as -4, -5 and so on? Because \(4^{-x} = \frac{1}{4^x}\). Am i missing something here?


When you take x as a negative integer, your both the statements A and B would not be satisfied.

Question clearly states that x<4, and both equations prove x=3.

Hence, Answer is D.

Note : try taking the negative values and substitute them in the equations, you will get the equations as incorrect. Hence, Negative values not allowed.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 24 May 2017, 23:27
My method of solving is similar to Bunuel in which we keep the solution terms of 4^x rather than 4^(x-1) as explained by VeritasPrepKarishma. 4^(x-1) maybe difficult for some people to understand but when you solve in 4^x, the expression is same as the one provided in question and makes reasoning and calculation much easier.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 24 Jun 2018, 20:18
1
Hi All,

We're told that X is an integer and 4^X < 100. We're asked for the value of X. This question can be solved by TESTing VALUES. There's also a great pattern behind the question that is asked. Since you're told that X is an INTEGER and 4^X < 100, that really means that X is less than or equal to 3.

1) 4^(X + 1) – 4^(X – 1) > 100

Fact 1 gives you a real ugly inequality, so instead of trying to do math, use the basic math that you already know and TEST VALUES...

IF....
If X = 2 ---> 64 - 4 = 60 which is NOT greater than 100, so x CANNOT = 2
If X = 3 ---> 256 - 16 = 240, which IS greater than 100.

There's no point in testing smaller integers, since the total won't be > 100 and you're NOT ALLOWED to go bigger than 3 (because of the information in the prompt). Therefore X must equal 3.
Fact 1 is SUFFICIENT

2) 4^(X + 1) + 4^X > 100

Fact 2 provides us with a similar situation, with just a slight change in the math.

IF....
If X = 2 ---> 64 + 16 = 80 which is NOT greater than 100, so X CANNOT = 2 (or anything smaller than 2)
If X = 3 ---> 256 + 64 = 320, which IS greater than 100, so X can be 3.
X can't be anything other than 3.
Fact 2 is SUFFICIENT.

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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 27 Jun 2018, 00:44
Can anyone please explain why cant we consider x=-4 or x=-5 in the first equation. this also satisfies the equation 4^x <100 and we get x=+3,-3,-4,-5........ So statement 1 is not sufficient. Please help.
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Re: If x is an integer and 4^x < 100, what is x?  [#permalink]

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New post 27 Jun 2018, 03:02
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ambujsrv wrote:
Can anyone please explain why cant we consider x=-4 or x=-5 in the first equation. this also satisfies the equation 4^x <100 and we get x=+3,-3,-4,-5........ So statement 1 is not sufficient. Please help.



(1) 4^(x + 1) – 4^(x – 1) > 100

x = -4 or -5 does not satisfy stmnt 1.

4^(-4 + 1) - 4^(-4-1)
= 4^(-3) - 4^(-5)
= 1/4^3 - 1/4^5

This is not greater than 100.

Only x = 3 satisfies this as well as 4^x < 100.
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Re: If x is an integer and 4^x < 100, what is x? &nbs [#permalink] 27 Jun 2018, 03:02
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