Find $t$ if: $\int_5^{\ln t} \ln x \, dx= \int_1^3 \ln x \, dx$ As the title suggests, I was asked by my friends this question today and had difficulty solving it.
$$\int_5^{\ln t} \ln x \, dx= \int_1^3 \ln x \, dx. $$
Integration by part tells us that the integral of $\ln x$ is $x\ln x-x.$ I am stuck after that.

Using $s=\log(t)$, you need to solve first for $s$ (this is what you already did properly)
$$s (\log(s)-1)=3 \log (3)+5 \log (5)-7$$ Since you are probably not supposed to know about Lambert function use inspection or, better, a plot to see that the solution is almost $s=6$. Zoom more and more to see that $s \sim 5.77$. Then, back to $t$.
What you could also do is to use a numerical method; this requires a reasonable starting point. Since the function shows a minimum at $s=1$, using Taylor
$$s (\log(s)-1)=-1+\frac{1}{2} (s-1)^2+O\left((s-1)^3\right)$$
Ignoring the higher order terms, the solution of the quadratic is
$$s_0=1+\sqrt{2 (-6+3 \log (3)+5 \log (5))}$$ Now, Newton iterates are
$$\left(
\begin{array}{cc}
n & s_n \\
 0 & 4.26895 \\
 1 & 5.93370 \\
 2 & 5.77134 \\
 3 & 5.77006
\end{array}
\right)$$
