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Stuck!

Here is a problem from the NMTC 2015 Bhaskara Contest:

\( \text{The number of values of}\space x\space \text{which satisfy the equation}\space 5^2.\sqrt[x]{8^{x-1}}=500\space \text{are:}\)

Here is the link. (Question 6).

I solved it like this:

\(5^2.\sqrt[x]{8^{x-1}}=500\)

\(\Rightarrow\space 5^2.(8^{x-1})^{\frac{1}{x}}=2^2\times5^3\)

\(\Rightarrow\space 5^2.8^{\frac{x-1}{x}}=(2^3)^{\frac{2}{3}}\times5^3\)

\(\Rightarrow\space 8^{\frac{x-1}{x}}=8^{\frac{2}{3}}\times5\)

But here, I am stuck. Any idea?

Note by Ashok Dargar
1 month, 1 week ago

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Actually, this is my father's account, and now I use it. I am in 7th standard. So please can you explain what 'log' is or explain without log?😊 Ashok Dargar · 1 month, 1 week ago

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@Ashok Dargar Check out the Brilliant wiki to learn logarithms which is pretty good. Sathvik Acharya · 1 month, 1 week ago

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There is no solution of this equation. if have any give that soln Ram Sita · 2 weeks, 4 days ago

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@Ram Sita It has no integral but a real sol Ram Sita · 2 weeks, 4 days ago

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@Ram Sita Yep you are right buddy! It has no integral solutions but one real solution. So the answer is 1 as the question does not specify whether they should be integers or real number. Sathvik Acharya · 2 weeks, 3 days ago

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Sir, you can continue from your approach by dividing both sides by \(8^{\frac{2}{3}}\) (in your final equation) and the applying log to the base \(10\) on both sides(A one step (log rules) simplification is required). This equation is obviously a linear equation, thus having only one solution. Sathvik Acharya · 1 month, 1 week ago

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