I am stuck solving this problem, any help please..Many thanks

Question

find all numbers x that satisfy the given equation ln(2x)
------- =2
ln(3x)

Answer 1

ln(2x)/ln(3x)=2
ln(2x)=2ln(3x)
ln(2x)=ln(9x²)
raise to power of e
e^ln(2x)=e^ln(9x²)
2x=9x²
x(9x-2)=0
x=0, or x=2/9
When working with log equations, always check back to see if the solution results in defined values.
Log(non-positive number) does not exist.

Check x=0:
Both log(2x) and log(3x) give undefined values, and
Lim x->0 log(2x)/log(3x)=1
and not 2 as required.

Check x=2/9:
log(2*x)/log(3*x)
=log(4/9)/log(6/9)
=log(4/9)/log(2/3)
=2log(2/3)/log(2/3)
=2 checks.

Answer 2

Thank you mathmate