Question

$a^2=lo{g}_{10}x,b^3=lo{g}_{10}y$ and $\frac{a^2}{2}-\frac{b^3}{3}=lo{g}_{{10}^z}$ express z in terms of x and y

Answer 1

$\frac{a^2}{2}-\frac{b^3}{3}=lo{g}_{10}z\Rightarrow \frac{{\log }_{10}x}{2}-\frac{{\log }_{10}y}{3}={\log }_{10}z\Rightarrow {\log }_{10}{x}^{1/2}-{\log }_{10}{y}^{1/3}={\log }_{10}z\Rightarrow {\log }_{10}\frac{x^{1/2}}{y^{1/3}}={\log }_{10}z$

Taking Antilog both sides

$\frac{x^{1/2}}{y^{1/3}}=z\Rightarrow [\frac{x^{1/2}}{y^{1/3}}{]}^6=z^6\Rightarrow \frac{x^3}{y^2}=z^6$