This is a homework question, but I've tried as hard as I can. Let me walk you through what I've done so far.
$\ln(x^2+1)+1 = \ln(x^2+4)$
$\ln(x^2+4) - \ln(x^2+1) = 1$
$\ln\left(\frac{x^2+4}{x^2+1}\right) = 1$
Now, this is where I'm kind of getting lost. Maybe I should rewrite the equation? Doesn't this basically say “e to the 1st power should be equal to $\frac{x^2+4}{x^2+1}$”?
$\frac{x^2+4}{x^2+1} = e$
$x^2+4 = ex^2+e$
This is where I couldn't move on, but as I was writing this post, this hit me:
$x^2-ex^2 = e-4$
$(1-e)x^2 = e-4$
$x^2 = \frac{e-4}{1-e}$
$x = \pm \sqrt{\frac{e-4}{1-e}}$
According to my book, the answer should be:
$x = \pm \sqrt{\frac{4-e}{e-1}}$
By calculating the right-hand expression, I see that it is the same as my answer.
$x \approx \pm 0.86$
Two questions:
- What's the reason for changing the order of terms in the solution?
- Have I made any particularly odd steps in my solution?