Algorithms-Time Complexity

Question

What is the time complexity of the below code?

for($k=n^{10};k \geq 5;k=k^{\frac{1}{7}},k=k^2$)

{  $k=k^5;$

    $k=k-10$

}

My answer comes to be $O(log_{\frac{7}{10}}log_5(n^{10}))$

Please verify.

Comments

yes its correct according to me..!

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@shubham and @ Ayush

Guys can you please post the detail solution as I'm not able to solve this question.

Thanks

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@mayanprakash

see $k$ is starting from $n^{10}$ and after one time execution of loop $k$ value is becoming $k=(((k^{5})^{1/7})^{2})= k^{\frac{10}{7}}$

so $(n^{10})^{(\frac{10}{7})^x}=5$ here $x$ will tell how many times will this loop execute until tue $k$ value becomes $5$

so take log on both side $(10/7)^{x} log_{5} n^{10}= 1$

after cross multiply $(7/10)^{x}=log_{5} n^{10}$

now take log again $x= log_ {\frac{7}{10}} log_{5} n^{10}$

its kind of just simplification..you need to do main part here is here how many times $k$ value is after one excn of loop and ignre small effects on $k$ value like here $k=k-10$ wont affect much so ignore it...addition and subtraction wont effect much so you can ignore it, just see for division and multiplication in this type of qsn..

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@Shubham-Please use Latex for more clear formulas.

Ref : https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference

 

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@ayush okk..   hvnt used till now will learn...thanks

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@Shubham @Ayush
$2^{nd} $ equation should be $(n^{10})^{(\frac{10}{7})^x}=5  \ or \ ((n^{10})^{\frac{10}{7}})^x=5$ (notice the power $x$).
Is it a typo?
Which one is correct?

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@soumya ...the thing you wrote first is correct

_actually in editing there is some mistake i will correct it..!

@srestha please correct it

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@shubham why r u doing equal to 5?

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@srestha

its because k value cant go below 5 as said in condn...so we are equating to 5 in order to know how mny times our loop will execute before its gets terminated.!

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@Shubham...
I am getting $2^{nd}$ one and a different answer :(
Each time we are taking $(\frac{10}{7})^{th}$ power of modified $k$ and that last value of k should be $\geq 5$. So shouldn't it be - $((n^{10})^{\frac{10}{7}})^x ?$

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@saumya how?

we are doing everytime k^10/7...if you apply two times than it will be k^(10/7^2)...similarly we are applying it for x times so k^((10/7)^x)..

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Ohh yes. Thank you @Shubham :)