Tangent graph to a circle I am trying to find the positive real number $b$ such that the graph of $f(x)=\ln(x+b)$ is tangent to the circle $x^2+y^2=1$. Here is my attempt:
I realize that the point of tangency must be in the top half of the circle if $b$ is positive, so I define $g(x)=\sqrt{1-x^2}$ (top half of circle). Now, I THINK that if two graphs $f$ and $g$ are tangent to each other at $x=a$, then $f(a)=g(a)$ and $f'(a)=g'(a)$ (correct me if I am wrong). Now, $f'(x)=\frac{1}{x+b}$, and $g'(x)=\frac{-x}{\sqrt{1-x^2}}$. So, we are basically solving for the $x$ value of the point of tangency and $b$. Now, I am not sure where to go from here. I used a graph to estimate a solution between $b=2.9$ and $b=3$. Can someone help?
 A: Solving the equation $f'(x)=g'(x)$ for $b$, you get
$$ b = -x - \frac{\sqrt{1-x^2}}{x} $$
Plug that in to $f(x) = g(x)$ and you have
$$ \ln \left(-\frac{\sqrt{1-x^2}}{x}\right) = \sqrt{1-x^2} $$
This is unlikely to have closed-form solutions, but you can use
numerical methods, such as Newton's.  Maple tells me
$$ x = -0.3669416757$$
which corresponds to 
$$ b = 2.902069222$$
A: Satrting from Robert Israel's answer, you look for the zero of function
$$f(x)=\ln \left(-\frac{\sqrt{1-x^2}}{x}\right) - \sqrt{1-x^2}$$ Plotting the function, we can see that the root is quite close to $-0.4=\frac 25$.
Peform a Taylor expansion around the value to get
$$f(x)=\left(\log
   \left(\frac{\sqrt{21}}{2}\right)-\frac{\sqrt{21}}{5}\right)+\left(\frac{12
   5}{42}-\frac{2}{\sqrt{21}}\right)
   \left(x+\frac{2}{5}\right)+\left(\frac{8125}{3528}+\frac{125}{42
   \sqrt{21}}\right)
   \left(x+\frac{2}{5}\right)^2+O\left(\left(x+\frac{2}{5}\right)^3\right)$$ which is a quadratic in $\left(x+\frac{2}{5}\right)$.
Solve it to get $x\approx -0.366863\implies b\approx 2.90262$ which is not too bad.
For sure, you can polish the root using Newton method and using the guess for $x_0$, the iterates would be
$$\left(
\begin{array}{cc}
 n & x_n \\
 0 & -0.36686326831355560408 \\
 1 & -0.36694166770061389757 \\
 2 & -0.36694167568391188355 \\
 3 & -0.36694167568391196631
\end{array}
\right)$$
