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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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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
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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
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^(x1) > 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
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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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
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. Answer: D.
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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
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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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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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+14^x1>100
so (15/4)*4^x>100
so 2xlg2>3lg2+1lg3, 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



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Re: If x is an integer and 4^x < 100, what is x? [#permalink]
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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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05 Nov 2015, 00:10
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 \(4^{x+1}  4^{x1}\) You take \(4^{x1}\) common because the exponent (x1) is smaller. \(4^{x1} ( 4*4  1)\) (because \(4^{x+1}\) has two extra 4s so they are left behind} \(4^{x1} * 15\)
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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. 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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03 Sep 2016, 02:29
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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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^(x1) as explained by VeritasPrepKarishma. 4^(x1) 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?
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