If $f\left(x\right)=\log x$ and $g\left(x\right)=x^3$ then $f\left(g\left(a\right)\right)+f\left(g\left(b\right)\right)$ is 

The given functions are $f\left(x\right)=\log x$ and $g\left(x\right)=x^3$

Consider $f\left(g\left(a\right)\right)+f\left(g\left(b\right)\right)$

$\begin{array}{rcl}f\left(g\left(a\right)\right)+f\left(g\left(b\right)\right)& =& \log \left(a^3\right)+\log \left(b^3\right)\\ & =& 3\log \left(a\right)+3\log \left(b\right)\\ & =& 3\left(\log \left(a\right)+\log \left(b\right)\right)\end{array}$

Use the logarithmic principle $\log x+\log y=\log xy$

The abvoe expression becomes

$\begin{array}{rcl}3\left\{\log \left(a\right)+\log \left(b\right)\right\}& =& 3\log \left(ab\right)\\ & =& \log \left({\left(ab\right)}^3\right)\\ & =& \boxed{f\left(g\left(ab\right)\right)}\end{array}$