Logarithms (logs) and exponents (indices) work together just as addition goes with subtraction and multiplication goes with division.
Your question: Why is
`log_(10)G = 2log_(10)M - log _(10)N` the same as`G=M^2/N`
relates to the rules of logs and exponents such as they relate to each other.
Now also remember that in exponents and logs plus (+) means multiplication and minus(-) means divide.
Therefore `log_(10)M - log_(10)N` is the same as the exponent written as `M/N` because the minus (-) means divide such as we have done by putting M over N.
Everything has the same "base" which is `log_(10)` and the `2log_(10)M` is the same as `log_(10)M^2`
As the bases are the same we can convert to an exponent :
`log_(10)G= log_(10)M^2 - log_(10)N`
`therefore G = M^2/N`