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If 2^x = 3, then which of the following must be true?

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Bunuel
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If 2^x = 3, then which of the following must be true? [#permalink] New post   06 May 2020, 10:31
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B
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If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

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Bunuel
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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   24 Nov 2021, 07:34
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Official Solution:

If \(2^x = 3\), then which of the following must be true?

A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)
B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)
C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)
D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)
E. \(1 \frac{5}{6} < x < 1\)


Square: \(2^{2x} = 3^2=9\);

\((2^{2x} = 9) > 2^3\), so \(2x > 3\);

\(x>1 \frac{1}{2}\)

Cube: \(2^{3x} = 3^3=27\);

\((2^{3x} = 27) < 2^5\), so \(3x < 5\);

\(x < 1 \frac{2}{3}\).

So, \(1 \frac{1}{2} < x < 1 \frac{2}{3}\).


Answer: B
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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   06 May 2020, 10:42
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Bunuel wrote:
If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

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We can do some manipulations to help us determine how big \(x\) is in \(2^x = 3\).

If we square both sides we can get \(2^{2x} = 3^2 = 9\). 9 is a little bit bigger than \(2^3\). So we have \(2^{2x} > 2^3\). It follows that \(2x > 3\) and \(x > \frac{3}{2}\).

We could choose B here since we know x should be just a little bit bigger than \(\frac{3}{2}\), but if we want to be safe we can try to find an upper bound for x.

Let's try using the fact that \(2^{3x} = 3^3 = 27\). We want an upper bound this time so we have \(2^{3x} < 32 = 2^5\) and \(3x < 5\), which is \(x < \frac{5}{3}\). Hence we can comfortably choose B.

Ans: B
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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   06 May 2020, 10:45
Answer is supposedly B.
A. 2^4/3= 2.51 < x < 2^3/2= 2.82 (less than 2^x=3)

B. 2^3/2 =2.82 < x < 2^5/3= 3.17 (3 lies in between)

C. 2^5/3= 3.17 < x < 2^7/4= 3.36 (More than 2^x=3)

D. 2^7/4= 3.36 < x < 2^11/6= 3.56 (More than 2^x=3)

E. 2^11/6= 3.56 < x < 2^2= 4 (More than 2^x=3)

B is the answer

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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   06 May 2020, 19:06
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\(x = log_2 3\)

\(x= \frac{log_{10}3}{log_{10}2}\)

\(x= \frac{0.47}{0.3} \)

\(1.6>\frac{0.47}{0.3}>1.5\)

OR

\(2^{1.5} = 2^{(1+0.5)} = 2*2^{0.5} = 2*1.414 = 2.82\)

\(2^{1+\frac{2}{3}} = 2*2^{\frac{2}{3}} = 2*4^{\frac{1}{3}} = 2*[\frac{500}{125}]^{\frac{1}{3}} \) ~ \(2*[\frac{512}{125}]^{\frac{1}{3}}\)= \(2*(\frac{8}{5}) = 3.2\)



Or

\((2^3)*(\frac{9}{8}) = 3^2\)

\(2^3*1.125 = 3^2\)

We know that 2^0= 1 and 2^1 = 2

1.125 must be in between \(2^{0.1}\) and \(2^{0.2}\)

\(2^3*2^{0.1} < 2^{2x} < 2^3*2^{0.2}\)

\(2^{3.1} < 2^{2x} < 2^{3.2}\)

\(2^{1.55}< 2^x< 2^{1.6}\)

B






Bunuel wrote:
If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   07 May 2020, 12:47
I solved this by "feeling". Let me explain. But I need experts to critic me on this method. nick1816 let me know your thoughts please?

So we know
2^1.1
2^1.2
2^1.3
.....
2^1.9
2^2

is going to be an exponential curve (NOT linear).

And if it was linear then 2^1.5 would be exactly 3, but its not.
So 2^1.5 < 3
But the rate of growth of this exponential curve is only slight (not as much as say 2^2, 2^3, 2^4...)
So 2^1.5 should be quite close to 3

I mean then I chose the option B which is closest. However, I do understand that this is not full proof (as we do not know how close 2^1.5 is to 3 unless we work it out)
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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   07 May 2020, 19:23
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You are on point bhai. Since the curve is exponential, 2^1.5 < 3.

Now you could find the approximate value of \(2^{(1\frac{2}{3})}\) just to make sure that you're marking the correct option. I've already mentioned one way to approximate \(2^{(1\frac{2}{3})}\)in my solution.

Another way is-

\(3375 < 4000 < 4096\)

\(15^3<4000<16^3\)

\(1.5^3 < 4 < 1.6^3\)

\(1.5 < 4^{(\frac{1}{3})} < 1.6\)

\(1.5 < 2^{(\frac{2}{3})} < 1.6\)

\(1.5*2 < 2*2^{(\frac{2}{3})} < 1.6*2\)

\(3 < 2^1*2^{(\frac{2}{3})} < 3.2\)

\(3 < 2^{(1+\frac{2}{3})} < 3.2\)



AnirudhaS wrote:
I solved this by "feeling". Let me explain. But I need experts to critic me on this method. nick1816 let me know your thoughts please?

So we know
2^1.1
2^1.2
2^1.3
.....
2^1.9
2^2

is going to be an exponential curve (NOT linear).

And if it was linear then 2^1.5 would be exactly 3, but its not.
So 2^1.5 < 3
But the rate of growth of this exponential curve is only slight (not as much as say 2^2, 2^3, 2^4...)
So 2^1.5 should be quite close to 3

I mean then I chose the option B which is closest. However, I do understand that this is not full proof (as we do not know how close 2^1.5 is to 3 unless we work it out)
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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   08 May 2020, 04:10
Bunuel wrote:
If \(2^x = 3\), then which of the following must be true?


A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)

B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)

C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)

D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)

E. \(1 \frac{5}{6} < x < 2\)

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2^x=3
2^x-2=1
2^(x-1)=3/2=1.5
\(\sqrt{2}=1.414\)
0.5<x<0.6
B:)
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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   10 Jul 2021, 15:05
Must admit I failed this question. But I found some valuable approximations of x. The fractions represent the closest distances between a power of 2 and a power of 3.

https://oeis.org/A254351/internal

"log(3)/log(2) = 1.5849625... is an irrational number. The fractions (2/1, 3/2, 5/3, 8/5, 11/7, 19/12, 46/29, 65/41, 84/53, 317/200, 401/253, 485/306, 569/359, 1054/665, ...) are a sequence of approximations to log(3)/log(2), where each is an improvement on its predecessors."

This means that 2^(19/12) is closer to 3 than 2^(11/7).

"Seven/eleven" seems easy to remember as a decent approximation. =)

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Re: If 2^x = 3, then which of the following must be true? [#permalink] New post   21 Apr 2024, 01:56
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Re: If 2^x = 3, then which of the following must be true? [#permalink]
21 Apr 2024, 01:56
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