Question 143215
<pre><font size = 4 color = "indigo"><b>
{{{ ln(x-1)-ln(x+1)=ln(x+5)}}}

Use this principle of logarithms:  {{{ln(A)-ln(B)=ln(A/B)}}}

to rewrite the left side;

{{{ln((x-1)/(x+1))=ln(x+5)}}}

Now use the principle that {{{ln(A)=ln(B)}}} can be written as {{{A=B}}}
to rewrite the whole equation:

{{{(x-1)/(x+1)=x+5}}}

Do you know how to solve that?  If not post again asking how.
You get

x=-3, x = -2

But these must check in the original problem.

{{{ ln(x-1)-ln(x+1)=ln(x+5)}}}

Substitute x=-3 in the original

{{{ ln(-3-1)-ln(-3+1)=ln(-3+5)}}}

{{{ ln(-4)-ln(-2)=ln(2)}}}

But there can be no logarithm of a negative number
(except in advanced mathematics, but never in ordinary algebra).

So -3 is NOT a solution

Substitute x=-2 in the original

{{{ ln(-2-1)-ln(-2+1)=ln(-2+5)}}}

{{{ ln(-3)-ln(-1)=ln(3)}}}

But, again, there can be no logarithm of a negative number
in ordinary algebra.

Therefore -2 is not a solution either.

Therefore there is no solution to the equation.

Edwin</pre>