If 3^x10, which of the following must be true?

Question

If 3^x > 10, which of the following must be true?

I. x > 2 
II. x > 3 
III. x > 4

A. I only 
B. II only 
C. III only 
D. I and II only 
E. I, II, and III

sashiim20 · Accepted Answer

3^x>10 3^2 = 9 3^3 = 27 >10 Therefore x must be 3 and greater. ie; x must be greater than 2 . x>2 ------- I Only Answer (A)... _________________ Please Press "+1 Kudos" to appreciate.

ScottTargetTestPrep · Answer

Since 3^2 = 9 and 3^3 = 27, if 3^x = 10, then x must be some number between 2 and 3. So if 3^x > 10, then x must be greater than 2. However, x may not need to be greater than 3 (or 4) to hold the inequality 3^x > 10. For example, if x = 2.5, 3^2.5 = 3^2 x 3^0.5 = 9√3 > 10.

Answer: A

Bunuel · Answer

3^x > 10 implies that x is greater than approximately 2.095, so x is definitely greater than 2. Whether x is an integer or not doesn't matter here, statement I must be true.

So, basically, if x = 2, then 3^x = 9, which is less than 10, so x must be greater than 2 for 3^x > 10 to hold true. Thus, statement I must be true. However, x could be exactly 3, in which case statements II and III are not necessarily true.

generis · Answer

habdo  , I agree with you. I toggled between answers A and E. I picked A on a gamble. Nothing more.

MathRevolution  , thank you for posting the question. It seems to me that by definition, if x > 2, it must also be greater than 3 and 4 unless there is an upper limit restriction, and here there is not.

I understand that Option I is the minimum condition which satisfies the inequality.

That is, x must be greater than 2 for  3^{x}  to be greater than 10.

But as I understand the word "must," in logic and in math, "must" includes the transitive cases:

If 3 > 2, and 2 > x, then 3 > x

I am not sure how one could argue that the third statement "must not" be true. Am I missing something?

Novice90 · Answer

As the question does not mention that x is an integer, x=2.01 then 3^2.01 = 9.09, which is not greater than 10. Hence x>10 cannot be the right answer.

Also, the question states that  "which ones of these must be true" . Any number with x>3 and x>4 will always be true, so why not select these options, as they supply the right numbers

SchruteDwight · Answer

ScottTargetTestPrep  by the same logic then,  x=2.01  would be greater than 2, however  3^x  would not (as not given that  x=integer )

ScottTargetTestPrep · Answer

The problem says given the premise that 3^x > 10, then x is... So x can't be 2.01 since 3^2.01 is not greater than 10. However, we know x is 2 point something. Now let's say that 2.1 is the smallest value such that 3^2.1 > 10. Thus, we cannot say that x must be greater than 3 (or 4), since 2.1, 2.2, 2.3, etc. are not greater than 3 (or 4), but it's true to say x must be greater than 2.

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Emily1122 · Answer

Bad question - it never said x is integer. X could be 2.01 which gives us 9.09942022744

sdas80923 · Answer

Since the question doesn't specify that x is an integer, should x>2 be incorrect? Since, x can then be 2.01 and 3^2.01 is not greater than 10.

Bunuel · Answer

Have you checked  this post ?

x cannot be 2.01 because it does not satisfy 3^x > 10.

3^x > 10 implies that x is greater than approximately 2.095, so x is definitely greater than 2.

This question belongs to a type of question that tends to be very confusing for many. Therefore, I recommend practicing similar questions from the following collection:  Trickiest Inequality Questions Type: Confusing Ranges .

Hope it's clear.

If 3^x>10, which of the following must be true?

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