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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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Updated on: 24 May 2017, 21:26
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54% (01:33) correct 46% (01:41) wrong based on 377 sessions

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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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16 Dec 2010, 05:08
2
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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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.

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

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16 Dec 2010, 05:18
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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16 Dec 2010, 05:25
1
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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16 Dec 2010, 05:40
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.

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

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16 Dec 2010, 06:27
3
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.

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Status: Bring the Rain Joined: 17 Aug 2010 Posts: 360 Location: United States (MD) Concentration: Strategy, Marketing Schools: Michigan (Ross) - Class of 2014 GMAT 1: 730 Q49 V39 GPA: 3.13 WE: Corporate Finance (Aerospace and Defense) Re: If x is an integer and 4^x < 100, what is x? [#permalink] ### Show Tags 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! _________________ Manager Joined: 13 Sep 2015 Posts: 83 Location: United States Concentration: Social Entrepreneurship, International Business GMAT 1: 770 Q50 V45 GPA: 3.84 Re: If x is an integer and 4^x < 100, what is x? [#permalink] ### Show Tags 01 Nov 2015, 13:12 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 D Intern Joined: 11 Oct 2015 Posts: 1 Re: If x is an integer and 4^x < 100, what is x? [#permalink] ### Show Tags 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. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8102 Location: Pune, India Re: If x is an integer and 4^x < 100, what is x? [#permalink] ### Show Tags 05 Nov 2015, 00:10 1 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} $$4^{x-1} * 15$$ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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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.

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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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.

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.

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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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.
Re: If x is an integer and 4^x < 100, what is x?   [#permalink] 24 May 2017, 23:27
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