ln(x-3)=ln(x+17)-ln(x-1)
1) ln A-ln B=ln (A/B)
ln(x-3)=ln[(x+17)/(x-1)]
Then:
x-3=(x+17)/(x-1)
(x-3)(x-1)=(x+17)
x²-x-3x+3=x+17
x²-5x-14=0
x=[5⁺₋√(25+56)]/2=(5⁺₋9)/2
We have two possible solutions:
x₁=(5-9)/2=-2
x₂=(5+9)/2=7
We must to check the possible solutions:
if x₁=-2; then; ln(-2-3)=ln(-2+17)-ln(-2-1), in this case this solutions is not possible, because logarithms of negative numbers are not defined.
if x₂=7; then:
ln(7-3)=ln(7+17)-ln(7-1)
ln 4=ln 24 -ln 6
ln 4=ln 24/6
ln 4=ln 4
Answer: x=7