Logarithm common factor problem

Question

there exist positive A, B, and C, with no common factor greater than 1, such that

$A.log_{200}5 +B.log_{200}2=C$

what is A + B +C

I dont know how to equal this equation

Answer 1

Take 200 to the power of each side. We get $5^A2^B=200^C$. Since $200=2^35^2$ we have $5^A2^B=5^{2C}2^{3C}$. Hence $A=2C,B=3C$. Since they have no common factor greater than 1, we must have $A=2,B=3,C=1$.