Log5(x-2)=3

If this is meant to be log to the base 5 of (x-2) is 3 then:

log₅(x-2)=3 means  x-2 = 5³, so x = 5³+2, or x=125+2, or x = 127.

 

If it is meant to be log to the base 10 of 5*(x-2) =3 then:

log₁₀(5*(x-2)) = 3 means  5*(x-2) = 10³, so x-2 = 200, so x = 202

 

If it is meant to be log to the base 10 of 5, all multiplied by (x-2) = 3 then:

(x-2)*log₁₀(5)=3, so x-2 = 3/log₁₀(5) so x = 3/log₁₀(5)+2 or $${\mathtt{x}} = {\frac{{\mathtt{3}}}{{log}_{10}\left({\mathtt{5}}\right)}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}} \Rightarrow {\mathtt{x}} = {\mathtt{6.292\: \!029\: \!674\: \!220\: \!178\: \!6}}$$