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

Analyzing the Logarithmic expression

Firstly, let’s understand how it is formed mathematically by analyzing the given logarithmic expression.

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

According to our analysis, we can identify following key factors in the expression.

1. The terms in the logarithmic expression are fractions basically. So, they cannot be added directly.
2. The logarithm of a number $30$ is common in the denominator of each term and it should be evaluated with respect to a base number.

Now, let’s prepare a plan of action to simplify the logarithmic expression and also to find its value.

Simplify the given Logarithmic expression

The reciprocal form of a logarithmic quantity interrupts us to add a log term to another term in this logarithmic expression. So, each term should be released from this form and it is possible in logarithms by the reciprocal rule of logarithms.

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

The reciprocal rule of logarithms changed the base of every logarithmic quantity in the expression. Now, all logarithmic terms have same base in the expression.

The sum of logarithmic terms can be calculated by the addition of logarithms.

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

$\implies$ $\log_{30}{2}$ $+$ $\log_{30}{3}$ $+$ $\log_{30}{5}$ $\,=\,$ $\log_{30}{(30)}$

The simplification process of the given logarithmic expression is completed successfully.

Find the value of simplified Log expression

Now, it is time to evaluate the simplified logarithmic expression the log of $30$ to base $30$.

$\,=\,\,\,$ $\log_{30}{30}$

The logarithm of $30$ to base $30$ is equal to one as per the log of base formula.

$\,\therefore\,\,\,$ $\log_{30}{30}$ $\,=\,$ $1$