We have that: $f'(x)=(a_1a_2...a_n)^x\ln(a_1a_2...a_n) - a_1^x\ln a_1-a_2^x\ln a_2-...-a_n^x\ln a_n=(a_1a_2...a_n)^x\ln a_1+(a_1a_2...a_n)^x\ln a_2+...+(a_1a_2...a_n)^x\ln a_n -a_1^x\ln a_1-a_2^x\ln a_2-...-a_n^x\ln a_n=a_1^x \ln a_1[(a_2...a_n)^x-1]+a_2^x\ln a_2[(a_1a_3...a_n)^x-1]+...+ a_n^x\ln a_n[(a_1a_2...a_{n-1})^x-1]$. We can conclude that $f'(x)>0$ because $a_1,a_2, \dots , a_n \in (1,\infty).$,hence $f$ is strictly increasing.