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If 0.845 is an approximate solution for the equation 10^x=7, which

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If 0.845 is an approximate solution for the equation 10^x=7, which  [#permalink]

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New post Updated on: 11 Mar 2019, 04:41
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GMATH practice exercise (Quant Class 3)

If 10^0.845 is approximately equal to 7, which of the following fractions is closest to the solution of the equation 7^(x+2) = 70?

(A) 29/211
(B) 19/107
(C) 31/169
(D) 43/211
(E) 53/211

P.S.: we thank kgmat12 for the question stem wording improvement!

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

Originally posted by fskilnik on 08 Mar 2019, 08:33.
Last edited by fskilnik on 11 Mar 2019, 04:41, edited 1 time in total.
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Re: If 0.845 is an approximate solution for the equation 10^x=7, which  [#permalink]

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New post 08 Mar 2019, 10:53
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\(10^x = 7\)

\(10 = 7^\frac{1}{x}\)

Now, let's look for the second one:

\(7^x * 7^2 = 7 * 10\)

\(7^x * 7 = 10\)

Compare:

\(x + 1 = \frac{1}{x}\)

Remember that \(\frac{1}{x} = \frac{1}{0.845}\)

Hence \(x = \frac{1 - 0.845}{0.845} = \frac{155}{845} = \frac{31}{169}\)

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Re: If 0.845 is an approximate solution for the equation 10^x=7, which  [#permalink]

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New post 08 Mar 2019, 11:21
fskilnik wrote:
GMATH practice exercise (Quant Class 3)

If 0.845 is an approximate solution for the equation 10^x = 7, which of the following fractions is closest to the solution of 7^(x+2) = 70?

(A) 29/211
(B) 19/107
(C) 31/169
(D) 43/211
(E) 53/211
Very nice, lucajava! Kudos and thanks for joining!

Let me present our official solution:

\({10^{0.845}} \cong 7\)

\({7^{x + 2}} = 70\)

\(?\,\, \cong x\)


\({7^{x + 2}} = 7 \cdot 10\,\,\,\,\, \Rightarrow \,\,\,\,\,{7^{x + 1}} = 10\,\,\,\,\, \Rightarrow \,\,\,\,\,{\left( {{7^{x + 1}}} \right)^{0.845}} = {10^{0.845}}\,\, \cong \,\,\,7\)

\({7^{0.845\left( {x + 1} \right)}} = {7^1}\,\,\,\, \Rightarrow \,\,\,\,\,x = {{1000} \over {845}} - 1 = {{155} \over {845}} = {{31} \over {169}}\,\,\,\, \Rightarrow \,\,\,\,\left( {\rm{C}} \right)\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If 0.845 is an approximate solution for the equation 10^x=7, which  [#permalink]

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New post 10 Mar 2019, 18:03
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The question is confusing. The same variable x is used twice and has two different values?
Should we expect Qs like this on the exam?
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Re: If 0.845 is an approximate solution for the equation 10^x=7, which  [#permalink]

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New post 10 Mar 2019, 18:16
kgmat12 wrote:
The question is confusing. The same variable x is used twice and has two different values?
Should we expect Qs like this on the exam?

Hi, kgmat12 !

Thinking twice, I believe you are right. I mean, although the question stem (seems to me) perfectly understandable ("x" is a dummy variable), I guess GMAT´s wording would be (better and) something like that:

If 10^0.845 is approximately equal to 7, which of the following fractions is closest to the solution of the equation 7^(x+2) = 70?

(I have changed the question stem of this problem in our course, also. Thanks and kudos for the contribution!)

Regards and success in your studies,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: If 0.845 is an approximate solution for the equation 10^x=7, which   [#permalink] 10 Mar 2019, 18:16
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