Answer
$t\approx -1.37765$
Work Step by Step
As $u(t)$ is given, this is the output value therefore we have to calculate the input value of $t$.
$u(t)=\frac{27.4}{1+13e^{2t}}$
$15=\frac{27.4}{1+13e^{2t}}$
$15\times(1+13e^{2t})=27.4$ ($1+13e^{2t}$ is not equal to 0, as any power of $e$ is positive)
$15+195e^{2t}=27.4$
$195e^{2t}=12.4$
$e^{2t}\approx0.06359$
$\ln(e^{2t})\approx \ln(0.06359)$ (We have to take the natural logarithm of both sides.)
$2t\approx -2.7553$
$t\approx -1.37765$
