Write as a single logarithmic

Question

1n(x^2-3x-40)-1n(x-8)

Answer 1

Simplifies to ln((x²-3x-40)/(x-8)).

x²-3x-40=(x-8)(x+5), so further simplification is ln(x+5).

However, the original expression cannot be evaluated for x=8 because ln(x-8) becomes ln(0) which cannot be evaluated. Also, x+5>0, x>-5 so that ln(x+5) can be evaluated.

x²-3x-40 must be greater than zero so that the log can be evaluated. Therefore x-8>0 and x+5>0⇒x>8; or x-8<0 and x+5<0⇒x<-5, which contradicts the requirement that ln(x+5) can be evaluated.

CONCLUSION

Simplification is ln(x+5) provided that x>8.