Logarithm:- Man-made and natural

$\begin{aligned} \log_{10\ln e^2} 400 & = \log_{10 × 2} 400\\ & = \log_{20} 400\\ & = \color{#69047E}{\boxed{2}} \end{aligned}$

For $\ce{LOG}_{(10 \ln e^2)}(400)$

$\begin{aligned} \ce{LOG}_{(10 \ln e^2)}(400) &= \ce{LOG}_{20}400 \quad\quad\quad\quad\quad{\ln e^2 =2 \space\text{then } 2\cdot 10 =20 } \\&= \ce{LOG}_{20}(20)^2 \\&=2\ce{LOG}_{20}(20) \\&=2 \space \square \end{aligned}$

$\large \text{ADIOS!!!}$

Nice problem.