If $\log \left(\frac{x+y}{3}\right)=\frac{1}{2}\log x+\frac{1}{2}\log y$, show that $\frac{x}{y}+\frac{y}{x}=7$.

 

If $\log \left(\frac{x-y}{4}\right)=\log \sqrt{x}+\log \sqrt{y}$, show that $(x+y{)}^2=20xy$

If $x={\log }_{a}bc, y={\log }_{b}ca, z={\log }_{c}ab$ then prove that $\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}=1$.

If $f\left(x\right)=3x+5, g\left(x\right)=6x-1$, and $(f+g) \left(x\right)=ax+b$, then find the value of $a+b$.

If $f\left(x\right)=3x+5, g\left(x\right)=6x-1$, then find $(f-g) \left(2\right)$.

If $f\left(x\right)=3x+5, g\left(x\right)=6x-1$, then find $\left(fg\right) \left(3\right)$.

If $f\left(x\right)=3x+5, g\left(x\right)=6x-1$, then find $\left(\frac{f}{g}\right) \left(x\right)$ and its domain.

Let $f:\{2, 4, 5\}\to \{2, 3, 6\}$ and $g:\{2, 3, 6\}\to \{2, 4\}$ be given by $f=\left\{(2, 3), (4, 6), (5, 2)\right\}$ and $g=\left\{\right(2, 4) (3, 4), (6, 2\left)\right\}$. Write down $g\circ f$

If $f\left(x\right)=2x^2+3, g\left(x\right)=5x-2$, then find $f\circ g$.