Please solve the following for x: ln 2x-ln(x-2)=ln 3?

ln(2x) = ln(3) + ln(x-2)

ln(2x) = ln(3*(x-2)) because the sum of ln A and ln B is ln AB

2x = 3(x-2) because A=B whenever ln A = ln B

2x = 3x-6

x = 6

ln(2x) - ln(x - 2) = ln(3)
ln[2x/(x - 2)] = ln(3)
2x/(x - 2) = 3
2x = 3(x - 2)
2x = 3x - 6
2x - 3x = -6
x = -6/(-1)
x = 6

ln2x-ln(x-2)=ln3
=> ln2x/(x-2) =ln3 (since lna - lnb = lna/b)
=> 2x/(x-2) = 3 (raising lhs & rhs to the power of e)
=>2x = 3x-6
=> x=6

Now, CHECK:
put x=6 in eqn. LHS = ln12 -ln4 =ln12/4 =ln3 =RHS
thus x=6 satisfies this eqn. Hence, answer is x=6

Then, ln 2x/(x-2)=ln 3
2x=3x-6
x=6
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Check ln 12 -ln 4 =ln 3

God bless you.