If $ \log_{5} x = k \cdot \log_{10} x$
find the value of $k$ (rough approximation) without using calculator.
what i did is:
$ \log_{5} x = \log_{5}10 \cdot \log_{10}x$
$\therefore \:k = \log_{5}10 $
$\Rightarrow\: (1+4)^k = 10 $
$ 10 = (1+4)^k= 1+ 4k + \frac{k(k-1)}{ 2!}k^2 + \frac{k(k-1)(k-2)}{3!}k^3\cdots$
but after that i could not find a way to get the value of k.
[would be appreciated if someone solves the above series for k.]
$$\frac{\ln x}{\ln 5}=\frac{k\ln x}{\ln 10}\implies k=\frac{\ln 10}{\ln 5}=1+\frac{\ln 2}{\ln 5}\approx 1.43$$ because $\frac{\ln 2}{\ln 5}\approx\frac37$, because $2^7\approx 5^3$.