# Solving a logarithmic equation

$\ln(x + 1) = 2 + \ln(x - 1)$; solve for $x$.

From there I get $$\ln \frac{x+1}{x-1} = 2.$$

Am I headed in the right direction, in our examples we would exponentiate both sides, does that still stand for this even though there's only a 2 on the right hand side?

-
Yes, you get $e^2$ on the right... – J. M. Dec 6 '11 at 16:12
Yes, you are doing well. After a short while you will get a linear equation for $x$. – André Nicolas Dec 6 '11 at 16:12
Yes, you are in the right direction. Now take exponentials on both sides to eliminate the $\ln$. – Arturo Magidin Dec 6 '11 at 16:13
Minor comment: The subtraction is fine of course, but myself I would prefer to take the exponential immediately, get $x+1=(e^2)(x-1)$. – André Nicolas Dec 6 '11 at 16:23

You are right. From

$$\ln \frac{x+1}{x-1} = 2.$$

we can write it as

$$e^{\ln \frac{x+1}{x-1}}=e^{2}$$

then

$$\frac{x+1}{x-1}=e^{2}$$

You can find the solution for $x$

-

Why not take the exponential of both sides immediately? We get $$x+1=(e^2)(x-1),$$ and it's over.

-
That would be a better way to do it. Most College Algebra texts I've seen have you "collect the logs" first. I suppose the authors want to give the student practice using the rules of logarithms. – David Mitra Dec 6 '11 at 16:39
That would be correct David, as we just went over the rules of logs the previous week. – erimar77 Dec 6 '11 at 16:44

Yes, you use the fact that $$\ln u=v\iff u=e^v.$$ So applying this to what you have (with $u={x+1\over x-1}$ and $v=2$): $$\tag{1}{x+1\over x-1}=e^2.$$ Multiplying both sides by $x-1$ gives $$\tag{2}x+1=e^2(x-1).$$ (note, here that $x=1$ is not a solution of (2); so (1) and (2) are equivalent equations)

Solving for $x$ in (2):

\eqalign{ &x+1= e^2 x-e^2\cr &\iff x-e^2x =-1-e^2\cr &\iff x(1-e^2)=-1-e^2\cr &\iff x= {-1-e^2\over 1-e^2} } Or $$x={e^2+1\over e^2-1}.$$

When solving logarithmic equations, you sould always check your answers. In particular, check that you don't wind up taking the logarithm of a non-positive quantity.

-