Answer
$$0.5217 x+0.7826 y-1.2375 z=-5.309$$
Work Step by Step
Given $$\ln \left[1+4 x^{2}+9 y^{4}\right]-0.1 z^{2}=0, \quad P=(3,1,6.1876)$$
Consider $f(x,y,z)=\ln \left[1+4 x^{2}+9 y^{4}\right]-0.1 z^{2} $, since
\begin{align*}
\nabla f&=\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle\\
&=\left\langle\frac{8 x}{1+4 x^{2}+9 y^{4}}, \frac{36 y^{3}}{1+4 x^{2}+9 y^{4}},-0.2 z\right)\\
\nabla f_{P}&=\langle -5, 9, 6\sqrt{e}\rangle
\end{align*}
Then the equation of the tangent plane at $P$ is given by
\begin{align*}
\nabla f_{P} \cdot\langle x-x_1, y-y_1, z-z_1\rangle&= 0\\
\langle 0.5217,0.7826,-1.2375\rangle\cdot\langle x-3, y-1, z-6.1876\rangle&=0\\
0.5217(x-3)+0.7826(y-1)-1.2375(z-6.1876)&=0\\
0.5217 x+0.7826 y-1.2375 z&=-5.309
\end{align*}
