Question

$log_{5}5 \sqrt{5 \sqrt{5 \sqrt{5.... \infty } } }$
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Answer 1

Topic :-

Logarithm

To Evaluate :-

$\sf{\log_{5}5\sqrt{5\sqrt{5\sqrt{5\cdots\cdots\infty}}}}$

Solution :-

$\sf{\log_{5}5\sqrt{5\sqrt{5\sqrt{5\cdots\cdots\infty}}}}$

$\sf{\log_{5}5\left(\sqrt{5\sqrt{5\sqrt{5\cdots\cdots\infty}}}\right)}$

$\sf{Let\:\sqrt{5\sqrt{5\sqrt{5\cdots\cdots\infty}}}=x}$

$\sf{\sqrt{5\left(\sqrt{5\sqrt{5\cdots\cdots\infty}}\right)}=x}$

$\sf{\sqrt{5(x)}=x}$

Squaring both sides,

$\sf{5x=x^2}$

$\sf{x^2-5x=0}$

$\sf{x(x-5)=0}$

$\sf{Either\:x=0\:or\:x=5}$

We reject x = 0, as square root of a positive number can't be Zero.

So, x = 5.

$\sf{\log_{5}5\left(\sqrt{5\sqrt{5\sqrt{5\cdots\cdots\infty}}}\right)=\log_{5}5(x)=\log_{5}5(5)}$

$\sf {\log_55(5)=\log_55^2}$

$\sf {\log_55^2=2\log_55}$

$\sf {(\because \log a^b=b\log a)}$

$\sf {2\log_55=2}$

$\sf {(\because \log_a a=1)}$

Answer :-

$\underline{\boxed{\sf{\log_{5}5\sqrt{5\sqrt{5\sqrt{5\cdots\cdots\infty}}}=2}}}$