If log10G = 2log10M – log10N, then `G=M^2/N`  .Explain the reason for this.       G = M2/N -  why? This is the answer but I am not sure why?

Expert Answers
durbanville eNotes educator| Certified Educator

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`