Moving parentheses vertically

Here you are, with pmatrix. I added a better-looking variant, with medium-sized exponents:

\documentclass{article}
\usepackage{amsmath, nccmath}

\begin{document}

\begin{align*}
& \text{solve}\raisebox{1.5ex}{$ \begin{pmatrix}10^{\displaystyle 0.78478 \cdot\left(\log \left(\frac{m}{173.961}\right)\right)^{\!2}}=1.2,m\end{pmatrix} $}\triangleright m=83.7164
\\[2ex]
 & \text{solve}\raisebox{1.05ex}{$ \begin{pmatrix}10^{\medmath{0.78478 \cdot\left(\log \left(\frac{m}{173.961}\right)\right)^{\!2}}}=1.2,m\end{pmatrix} $}\triangleright m=83.7164
\end{align*}

\end{document} 

The brackets should be centred on the math axis, raising them makes the expression look distorted and hard to read. The apparent extra space is simply due to the brackets being a lot bigger than necessary.

I would use one of the following layouts

\documentclass{article}
\usepackage{amsmath,amssymb}

\begin{document}
normal
\[
\operatorname{solve}
\bigl(10^{0.78478 \cdot
(\log \frac{m}{173.961})^{2}}
=1.2,m\bigr) \triangleright m=83.7164
\]

bigger fraction
\[
\operatorname{solve}
\bigl(10^{0.78478 \cdot
\bigl(\log \tfrac{m}{173.961}\bigr)^{\!2}}
=1.2,m\bigr) \triangleright m=83.7164
\]
\end{document}

Also note that you should not use \text for a function name, it has the wrong spacing and uses the current text font so for example it would be italic in a theorem.

Here's a solution that doesn't raise the exponent term above the baseline. Just make sure to explain to your readers that ^ (generated by \widehat{\phantom{t}}) is the exponentiation operator.

\documentclass{article}
\usepackage{amsmath,amssymb,mleftright}
\mleftright
\begin{document}
\[
\text{solve }
\left(10 \widehat{\phantom{t}} \mkern-1.5mu
\left[ 0.78478 \cdot
\left(\log \frac{m}{173.961}\right)^{2}\right]
=1.2,m\right) \triangleright m=83.7164
\]
\end{document}