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Evaluate $\dfrac{1}{\log_{2} 30}$ $+$ $\dfrac{1}{\log_{3} 30}$ $+$ $\dfrac{1}{\log_{5} 30}$

Logarithm of $30$ to base $2$, base $3$ and base $5$ are involved in reciprocal form and three logarithmic terms are connected each other by the summation.

$\dfrac{1}{\log_{2} 30}$ $+$ $\dfrac{1}{\log_{3} 30}$ $+$ $\dfrac{1}{\log_{5} 30}$

In this logarithmic problem, the bases are different but the number is same. If the reciprocal form of each logarithm term is reversed, then all three log terms have the common base. So, use change base rule of logarithm in reciprocal form to simplify this logarithmic expression.

$= \log_{30} 2$ $+$ $\log_{30} 3$ $+$ $\log_{30} 5$

Now, all three terms are having same base and they are connected in summation form. So, use product rule of logarithms to express sum of logarithms of all three terms in product form.

$= \log_{30} (2 \times 3 \times 5)$

$= \log_{30} (30)$

As per logarithm of a number to same number rule, the logarithm of number $30$ to base $30$ is one.

$\implies \log_{30} 30 = 1$

$\therefore \,\,\,\,\,\, \dfrac{1}{\log_{2} 30}$ $+$ $\dfrac{1}{\log_{3} 30}$ $+$ $\dfrac{1}{\log_{5} 30} = 1$

Therefore, it is proved that the summation of the reciprocals of logarithm of number $30$ to bases $2$, $3$, and $5$ is one and it is the required solution for this logarithmic problem in mathematics.

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