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Find the value of $\log_{3} (\log x^3)$ $-$$\log_{3} (\log x)$

$\log_{3} (\log x^3) -\log_{3} (\log x)$

Two logarithmic terms are formed an expression in subtraction form and they contain $3$ as a base commonly but each logarithmic function contains another logarithmic function internally. The value of subtraction of them is required to find in this logarithm problem.

The logarithmic problem can be solved in two different methods possibly.

Method: 1

$\log_{3} (\log x^3) -\log_{3} (\log x)$

Step: 1

The logarithmic functions have same base and it is $3$. One logarithmic function is subtracted from another function and they can be simplified by applying quotient rule of logarithms.

$=\,\,$ $\log_{3} \Bigg[\dfrac{\log x^3}{\log x}\Bigg]$

Step: 2

Consider $\log x^3$ term and it is a logarithm of an exponential term. It can be expressed in another form by using power law of logarithm.

$=\,\,$ $\log_{3} \Bigg[\dfrac{3 \times \log x}{\log x}\Bigg]$

$=\,\,$ $\log_{3} \Bigg[3 \times \dfrac{\log x}{\log x}\Bigg]$

$=\,\,$ $\require{cancel} \log_{3} \Bigg[3 \times \dfrac{\cancel{\log x}}{\cancel{\log x}}\Bigg]$

$=\,\,$ $\log_{3} (3 \times 1)$

Step: 3

$=\,\,$ $\log_{3} 3$

The logarithm of a number to same number is always one. Therefore, the logarithm of $3$ to $3$ is one.

$\therefore \,\,\,\,\,\, \log_{3} (\log x^3) -\log_{3} (\log x) = 1$

It can also be solved in another method of the logarithms.

Method: 2

$\log_{3} (\log x^3) -\log_{3} (\log x)$

Step: 1

The first logarithmic term contains another logarithm term but it contains an exponential term. Simplify it by using power rule formula of logarithms.

$\log_{3} (3 \times \log x) -\log_{3} (\log x)$

Step: 2

Inside the first logarithm term, two numbers are multiplying each other and it represents the product rule of logarithm. Use it and split the first logarithmic function as the sum of two logarithmic terms.

$=\,\,$ $\log_{3} 3 + \log_{3} (\log x) -\log_{3} (\log x)$

$=\,\,$ $\require{cancel} \log_{3} 3 + \cancel{\log_{3} (\log x)} -\cancel{\log_{3} (\log x)}$

$=\,\,$ $\log_{3} 3$

Step: 3

The value of logarithm of $3$ to $3$ is one.

$\therefore \,\,\,\,\,\, \log_{3} (\log x^3) -\log_{3} (\log x) = 1$

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