= Solution
Assume first that $T>0$ and $-1<b<r<g$, so the <logarithmic utility> is defined throughout the allowed <portfolio> interval. Put $A=g-r>0$ and $B=r-b>0$. Then
$$
U'(x)=-\frac{pA}{T(1+g)-Ax}+\frac{(1-p)B}{T(1+b)+Bx}.
$$
Equating this to zero and clearing the positive denominators gives
$$
x_0=T\left[\frac{(1-p)(1+g)}{g-r}-\frac{p(1+b)}{r-b}\right].
$$
The second <derivative> is strictly negative, and $U'(T)<0$ by part (c), so $x_0<T$. The <two-state logarithmic portfolio with a borrowing constraint> is therefore
$$
\boxed{x_*=\max\{0,x_0\},\qquad\text{equity investment}=T-x_*.}
$$
In particular, the threshold for all equity is
$$
p\ge p_c:=\frac{(r-b)(1+g)}{(r-b)(1+g)+(g-r)(1+b)}
=\frac{(r-b)(1+g)}{(g-b)(1+r)}.
$$
For $p<p_c$ the optimum is interior. The probability bound printed in the PDF has $1+r$ where the all-equity threshold has $1+b$. Since $1+r>1+b$, that printed lower bound is strictly below $p_c$ and does not determine which of these two cases occurs. For example, with $(b,r,g)=(0,0.1,0.3)$ its lower bound is $13/35$, while $p_c=13/33$. At $p=0.38$ all the given inequalities hold and $x_*=0.23T$; at $p=0.4$ they also hold and $x_*=0$. Thus the boxed constrained formula applies to the printed data without replacing its probability assumption.
If bad-state gross wealth can be nonpositive, the <logarithmic utility> domain must instead be imposed explicitly. With $1+r>0$, $b\le-1$, and $0<p<1$, admissibility requires $x>-T(1+b)/(r-b)$; the same stationary root lies inside this positive-wealth interval and is optimal. If even the fully safe gross return is nonpositive, there is no admissible positive-wealth <portfolio>. These domain issues are implicit in using $\log w$.
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