The Laplace transform $u( {x;s;\varepsilon } )$ of the first passage time density $f( {t,x} )$ for a nearly logistic population under small demographic stochasticity satisfies a second-order differential equation \[ \frac{\varepsilon }{2}xp( {x;\varepsilon } )u_{xx} + x( {a - x} )q( {x;\varepsilon } )u_x - su = 0, \] on an interval $( {0,b} )$ with $0 < \varepsilon \ll 1$. For sufficiently small $\varepsilon $, the point $x = a$ is a second-order turning point in the interior, $0 < a < b$. The boundary point $x = 0$ is a regular singular point. The parameter s is the Laplace variable. Such problems as this ordinary differential equation (ODE) arise in modeling population growth and extinction and in genetics. The matched uniform asymptotic approximations found by Wazwaz and Hanson [SIAM J. Appl. Math., 46 (1986), pp. 943–961.] are applied to the first passage time problem with an absorbing boundary condition at $x = 0$, and a reflecting boundary condition at $x = b$. These results were employed to construct the coefficients of the exponentially dominant terms for the moments of extinction time to first order in $\varepsilon $. The mean and variance of the first passage time are found to be exponentially large as $\varepsilon \to 0 + $. The singular point results are essentially uniform on the full interval $( {0,b} )$. Exponential precision asymptotics yields the coefficient of the exponentially dominant term for the full extinction time moment generating function to first order in $\varepsilon $, and shows that extinction time distribution is exponential outside of the singular boundary layer, while inside the boundary layer it is the sum of an exponential distribution and a discrete instantaneous absorption distribution. For this application, the results are not accessible by the usual algebraic asymptotic approximations alone, and algebraic asymptotics do not play a crucial role in the final asymptotic analysis.