Question

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. ln (x - 6) + ln (x + 1) = ln (x - 15)

Answer 1

$\bf \textit{logarithm of factors}\\\\ log_{{ a}}(xy)\implies log_{{ a}}(x)+log_{{ a}}(y)\\\\ -------------------------------\\\\ ln(x-6)+ln(x+10)=ln(x-15) \\\\\\ ln[(x-6)(x+10)]=ln(x-15)\implies (x-6)(x+10)=x-15 \\\\\\ x^2-5x-6=x-15\implies x^2-6x+9=0 \\\\\\ (x-3)^2=0\implies \boxed{x=3}$

ok.. now, if we use  that on say ln(x-15), we end up with   $\bf ln(-12)\implies log_e(-12)$.

now, there's no exponent whatsoever, that would give us a negative result for a base of "e", namely, x = 3 is NOT in the domain for ln(), so the system is inconsistent, or has no solution.