Question

If $a^x=b^y=c^z=d^w$ then the value of $x\left[\right(1/y)+(1/z)+(1/w\left)\right]$is

Answer 1

Finding the value of $x\left[\right(1/y)+(1/z)+(1/w\left)\right]$:

Given $a^x=b^y=c^z=d^w$

$$\begin{gathered} a^x=b^y=c^z=d^w \\ \log ax=\log by \\ x\log a=y\log b \\ (x/y)=[\log b]/[\log a]={\log }_{a}b \\ {\log }_{a}b=x/y \\ {\log }_{a}c=x/z \\ {\log }_{a}d=x/w \\ {\log }_a\left(bcd\right)={\log }_{a}b+{\log }_{a}c+{\log }_{a}d \\ =(x/y)+(x/z)+(x/w) \\ =x\left[\right(1/y)+(1/z)+(1/w\left)\right] \\ {\log }_a\left(bcd\right)=x\left[\right(1/y)+(1/z)+1/w)] \end{gathered}$$

Hence, the value of $x\left[\right(1/y)+(1/z)+(1/w\left)\right]$is ${\log }_a\left(bcd\right)$.