Question

If $log(2^a\times 3^b\times 5^c)$ is the arithmetic mean of $log(2^2\times 3^3\times 5),log(2^6\times 3\times 5^7),andlog(2\times 3^2\times 5^4),$ then a equals ___

Answer 1

$log(2^a+3^b+5^c)=\left[log\right(2^2\times 3^3\times 5)+log(2^6\times 3\times 5^7)+log(2\times 3^2\times 5^4\left)\right]/3\Rightarrow log(2^a\times 3^b\times 5^c{)}^3=log(2^2\times 3^3\times 5^1)+log(2^6\times 3^1\times 5^7)+log(2^1\times 3^2\times 5^4)\Rightarrow log(2^a\times 3^b\times 5^c{)}^3=log(2^2\times 3^3\times 5^1\times 2^6\times 3^1\times 5^7\times 2^1\times 3^2\times 5^4)\Rightarrow 2^{3a}\times 3^{3b}\times 5^{3c}=2^9\times 3^6\times 5^{12}$
Equating powers of 2, we get 3a = 9
$\Rightarrow$ a = 3.