Question

Evaluate:
$\log_{5} \frac{\sqrt[4]{25}}{625}$

Answer 1

$\underline{\underline{\huge{\textbf{Solution:}}}}$

$\log_{5} \frac{\sqrt[4]{25}}{625}$

$\underline{\huge{\textsf{Step-1:}}}$
Express 25 and 625 in terms of same base (i.e., 5)

$\log_{5} \frac{\sqrt[4]{5^{2}}}{5^{4}}$

$\implies \log_{5} \dfrac{(5)^{2(\frac{1}{4})}}{5^{4}}$

$\underline{\huge{\textsf{Step-2:}}}$
Using the Identity $(a^{m})^{n} = a^{mn}$

$\implies {5}^{2(\frac{1}{4})} = 5^{(\frac{1}{2})}$

$\implies \log_{5} \dfrac{5^{(\frac{1}{2})}}{5^{4}}$

$\underline{\huge{\textsf{Step-3:}}}$
Using the Identity $\dfrac{a^{m}}{a^{n}} = a^{m-n}$

$\implies \frac{5^{(\frac{1}{2})}}{5^{4}} = 5^{(\frac{1}{2})-4}$

$\implies \frac{5^{(\frac{1}{2})}}{5^{2}} = 5^{(\frac{1-8}{2})}$

$\implies \frac{5^{(\frac{1}{2})}}{5^{2}} = 5^{(\frac{-7}{2})}$

$\implies \log_{5} 5^{(\frac{-7}{2})}$

$\underline{\huge{\textsf{Step-4:}}}$
Using the Property of Logarithm $\log_{a} a^{x} = x\: \forall \, a>0$

$\implies \log_{5} 5^{(\frac{-7}{2})} = (\frac{-7}{2})}$

$\therefore$

$\blacksquare \: \: \mathbf{\log_{5} \dfrac{\sqrt[4]{25}}{625} = \underline{(\frac{-7}{2})}}$