Question

If $x=\log \frac{3}{5},y=\log \frac{5}{4}$ and $z=2\log \frac{\sqrt{3}}{2}$, prove that $5^{x+y-z}=1$

Answer 1

$x+y-z=\log \frac{3}{5}+\log \frac{5}{4}-2\log \frac{\sqrt{3}}{2}$

$=\log \frac{3}{5}+\frac{5}{4}-\log {\left(\frac{\sqrt{3}}{2}\right)}^2$

$=\log \frac{3}{4}-\log \frac{3}{4}=0$

$\therefore 5^{x+y-z}=5^{\circ }=1$

Hence proved