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?

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

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?😊

Aaryan Maheshwari - 3 years, 11 months ago

Check out the Brilliant wiki to learn logarithms which is pretty good.

Sathvik Acharya - 3 years, 11 months ago

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 - 3 years, 11 months ago

There is no solution of this equation. if have any give that soln

Ram Sita - 3 years, 10 months ago

It has no integral but a real sol

Ram Sita - 3 years, 10 months ago

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 - 3 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$