The sum of logarithm of $x$ to base $5$ and logarithm of $5$ to base $x$ is equal to quotient of $5$ by $2$. This logarithmic equation helps us to find the value of $x$. It can be solved in mathematics by using logarithmic system.

$\log_{5} x + \log_{x} 5 = \dfrac{5}{2}$

In this problem, logarithm of numbers and bases in two terms of the expression are in reciprocal form. So, it is not easy to solve this problem. However, the obstacle can be overcome by expressing both terms in reciprocal form by applying the reciprocal form of change of base rule of logarithm.

$\implies \log_{5} x + \dfrac{1}{\log_{5} x} = \dfrac{5}{2}$

Now, the both logarithmic terms contain same numbers and bases.

In order to solve this equation easily, take $y = \log_{5} x$. Now, transform the entire logarithmic expression in terms of $y$ and then start simplifying it.

$\implies y + \dfrac{1}{y} = \dfrac{5}{2}$

$\implies \dfrac{y^2+1}{y} = \dfrac{5}{2}$

Use cross multiplication method and simplify it further.

$\implies 2(y^2+1) = 5y$

$\implies 2y^2+2-5y=0$

$\implies 2y^2-5y+2=0$

It is a quadratic equation and it can be solved in mathematics by using quadratic formula method.

$\implies y = \dfrac{-(-5) \pm \sqrt{{(-5)}^2-4 \times 2 \times 2}}{2 \times 2}$

$\implies y = \dfrac{5 \pm \sqrt{25-16}}{4}$

$\implies y = \dfrac{5 \pm \sqrt{9}}{4}$

$\implies y = \dfrac{5 \pm 3}{4}$

$\implies y = \dfrac{5+3}{4}$ and $y = \dfrac{5-3}{4}$

$\implies y = \dfrac{8}{4}$ and $y = \dfrac{2}{4}$

$\therefore \,\,\,\,\,\, y = 2$ and $y = \dfrac{1}{2}$

Therefore, the values of $y$ are $2$ and $\dfrac{1}{2}$.

Actually the value of $y$ is $\log_{5} x$ as per our assumption.

Therefore, $\log_{5} x = 2$ and $\log_{5} x = \dfrac{1}{2}$.

As per $\log_{5} x = 2$ logarithmic equation.

$\log_{5} x = 2 \Leftrightarrow x = 5^2$

$\implies x = 25$

As per $\log_{5} x = \dfrac{1}{2}$ logarithmic equation.

$\log_{5} x = \dfrac{1}{2} \Leftrightarrow x = {(5)}^{\dfrac{1}{2}}$

$\implies x = \sqrt{5}$

Therefore, the values of $x$ are $25$ and $\sqrt{5}$. It is the required solution for this logarithmic problem mathematically.

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