Algebra problem No.1

Given that $\displaystyle 2^{x}3^{y}5^{z} = 2520 \implies x\ln2+y\ln3+z\ln5+t\ln7=\ln2520$ .

By Cauchy-Schwarz inequality , we have:

$(\displaystyle \ln^{2} 2 + \ln^{2} 3 + \ln^{2} 5 + \ln^{2} 7)(x^{2} +y^{2} +z^{2} +t^{2} \geq x\ln2+y\ln3+z\ln5+t\ln7)^{2} = \ln^{2} 2520$

$\displaystyle ⟹ (x^{2} +y^{2} +z^{2} +t^{2}) \geq \frac{\ln^{2} 2520}{\ln^{2} 2 + \ln^{2} 3 + \ln^{2} 5 + \ln^{2} 7} \approx \boxed{7.606458148...}$ .

Equality occurs when:

$\displaystyle \frac {x}{\ln 2} = \frac {y}{\ln 3} = \frac {z}{\ln 5} = \frac {t}{\ln 7}$ .