So Many Logs!

Algebra Level 4

6log5xlog3(x5)5log66xlog3(x3)=6log55x5log6x \large { 6 }^{ \log _{ 5 }{ x } }\log _{ 3 }( { x }^{ 5 } ) -{ 5 }^{ \log _{ 6 }{ 6x } }\log _{ 3 }{ \left( \dfrac { x }{ 3 }\right) } ={ 6 }^{ \log _{ 5 }{ 5x } }-{ 5 }^{ \log _{ 6 }{ x } }

If the sum of the solutions to the equation above is equal to abc+d,\large { a }^{ \frac { b }{ c } }+d,

where a,b,ca, b, c and dd are all positive integers with bb and cc are coprime, what is the smallest possible value of abc+dabc+d?

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