Question

Show that xyz=2. If $$x={\log }_{7}9$$, $y={\log }_{5}7$, $z={\log }_{3}5$.

Answer 1

Hint: For solving this question first we will write $x$, $y$ and $z$ into fraction form with the help of some logarithmic identities. Then we will multiply them easily to prove the result.

Complete step-by-step answer:
Given:
It is given that if $$x={\log }_{7}9$$ , $y={\log }_{5}7$, $z={\log }_{3}5$ and we have to prove that $xyz=2$.
Now, before we proceed we should be familiar with the following logarithmic formulas:
1. If $a$ and $b$ are two positive numbers then, ${\log }_{b}a={\displaystyle \frac{{\log }_{10}a}{{\log }_{10}b}}={\displaystyle \frac{\log a}{\log b}}$.
2. If $a$ is any positive number then, $\log a^n=n\log a$.
Now, we will use the above-mentioned formulas to write $x$, $y$ and $z$ into some other form that will help us to find the value of $xyz$ easily.
Now, as it is given that $$x={\log }_{7}9$$ , using the formula mentioned in the first and second point. Then,
$$\begin{array}{rl}& x={\log }_{7}9\\ & \Rightarrow x={\displaystyle \frac{\log 9}{\log 7}}={\displaystyle \frac{\log 3^2}{\log 7}}\\ & \Rightarrow x={\displaystyle \frac{2\log 3}{\log 7}}..................\left(1\right)\end{array}$$
Now, as it is given that $y={\log }_{5}7$ , using the formula mentioned in the first point. Then,
$\begin{array}{rl}& y={\log }_{5}7\\ & \Rightarrow y={\displaystyle \frac{\log 7}{\log 5}}..............\left(2\right)\end{array}$
Now, as it is given that $z={\log }_{3}5$ , using the formula mentioned in the first point. Then,
$\begin{array}{rl}& z={\log }_{3}5\\ & \Rightarrow z={\displaystyle \frac{\log 5}{\log 3}}................\left(3\right)\end{array}$
Now, from equation (1) we have $$x={\displaystyle \frac{2\log 3}{\log 7}}$$ , from equation (2) we have $y={\displaystyle \frac{\log 7}{\log 5}}$ and from equation (3) we have $z={\displaystyle \frac{\log 5}{\log 3}}$ . Then,
$\begin{array}{rl}& xyz={\displaystyle \frac{2\log 3}{\log 7}}\times {\displaystyle \frac{\log 7}{\log 5}}\times {\displaystyle \frac{\log 5}{\log 3}}\\ & \Rightarrow xyz=2\end{array}$

Thus, the value of $xyz=2$.
Hence, proved.

Note: Here, the student should avoid multiplying the given terms directly without using logarithmic identities. Firstly, the value of given terms should be written in such a manner so that when we multiply them there won’t be any difficulty. Moreover, students should avoid calculation mistakes while solving the problem to get the correct answer quickly.