Question

given that log(base 5)x=y, express log(base 25)x in terms of y

Answer 1

\[\Large \log_{5}(x) = y\]


\[\Large \frac{\log_{10}(x)}{\log_{10}(5)} = y\]


\[\Large \log_{10}(x) = y\log_{10}(5)\]


\[\Large 2\log_{10}(x) = 2y\log_{10}(5)\]


\[\Large 2\log_{10}(x) = y\log_{10}(5^2)\]


\[\Large 2\log_{10}(x) = y\log_{10}(25)\]


\[\Large 2\frac{\log_{10}(x)}{\log_{10}(25)} = y\]


\[\Large 2\log_{25}(x) = y\]


\[\Large \log_{25}(x) = \frac{y}{2}\]

Answer 2

or we could approach it this way, without changing the base
\[\log_5(x)=y\iff 5^y=x\iff 25^{\frac{y}{2}}=x\iff \log_{25}(x)=\frac{y}{2}\]