Solve for x

ln (x-3) = ln (x+17) - ln (x-1)

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