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Hint: Write $\dfrac{1}{256}$ in the form of ${{2}^{n}}$ and also change the decimal form of 0.3 into fractional form. Then use properties of logarithms to find the required value. Use base to power conversion formula of logarithms.

__Complete step-by-step answer:__

First we should understand the term â€˜logarithmâ€™ and then we will see the property of logarithm required to solve this question. In mathematics, the logarithm is the inverse function of exponentiation. That means that the logarithm of a given number â€˜nâ€™ is the exponent to which another fixed number the base â€˜bâ€™ must be raised, to produce that number â€˜nâ€™. Common logarithm has base 10, however we can convert it to any number. Let us take an example: consider a number, here I am using 100, so, 100 can be written as 10 raised to the power 2 or mathematically, ${{10}^{2}}$. Now, we have to find the logarithmic value of 100 with 10 as considering the base of the logarithm. In other words, we can interpret the question as â€˜to how much must be the power of 10 should be raised, so that it becomes equal to 100â€™. We know that 10 raised to power 2 is equal to 100, so the answer is 2. Mathematically, it can be written as ${{\log }_{10}}100=2$. Some important formulas for logarithms are:

$\begin{align}

& {{\log }_{m}}{{n}^{a}}=a{{\log }_{m}}n,\text{ } \\

& {{\log }_{a}}\left( m\times n \right)={{\log }_{a}}m+{{\log }_{a}}n\text{ } \\

& \text{lo}{{\text{g}}_{a}}\left( \dfrac{m}{n} \right)={{\log }_{a}}m-{{\log }_{a}}n \\

& {{\log }_{{{a}^{b}}}}m=\dfrac{1}{b}{{\log }_{a}}m \\

\end{align}$

Now, come to the question, $\dfrac{1}{256}$ can be written as $\dfrac{1}{{{2}^{8}}}={{2}^{-8}}$.

Now, ${{\log }_{2}}{{2}^{-8}}=(-8\times {{\log }_{2}}2)=(-8\times 1)=-8$.

Also, $0.3=\dfrac{3}{10}$. Therefore, ${{\log }_{9}}0.3={{\log }_{9}}\left( \dfrac{3}{10} \right)={{\log }_{{{3}^{2}}}}\left( \dfrac{3}{10} \right)=\dfrac{1}{2}\left( {{\log }_{3}}3-{{\log }_{3}}10 \right)=\dfrac{1}{2}\left( 1-{{\log }_{3}}10 \right)$.

Note: we would not have easily solved the question if we would not have converted $\dfrac{1}{256}$ into exponent form. Log 10 to the base 3 can be calculated by using a calculator. Fractional conversion of 0.3 was necessary.

First we should understand the term â€˜logarithmâ€™ and then we will see the property of logarithm required to solve this question. In mathematics, the logarithm is the inverse function of exponentiation. That means that the logarithm of a given number â€˜nâ€™ is the exponent to which another fixed number the base â€˜bâ€™ must be raised, to produce that number â€˜nâ€™. Common logarithm has base 10, however we can convert it to any number. Let us take an example: consider a number, here I am using 100, so, 100 can be written as 10 raised to the power 2 or mathematically, ${{10}^{2}}$. Now, we have to find the logarithmic value of 100 with 10 as considering the base of the logarithm. In other words, we can interpret the question as â€˜to how much must be the power of 10 should be raised, so that it becomes equal to 100â€™. We know that 10 raised to power 2 is equal to 100, so the answer is 2. Mathematically, it can be written as ${{\log }_{10}}100=2$. Some important formulas for logarithms are:

$\begin{align}

& {{\log }_{m}}{{n}^{a}}=a{{\log }_{m}}n,\text{ } \\

& {{\log }_{a}}\left( m\times n \right)={{\log }_{a}}m+{{\log }_{a}}n\text{ } \\

& \text{lo}{{\text{g}}_{a}}\left( \dfrac{m}{n} \right)={{\log }_{a}}m-{{\log }_{a}}n \\

& {{\log }_{{{a}^{b}}}}m=\dfrac{1}{b}{{\log }_{a}}m \\

\end{align}$

Now, come to the question, $\dfrac{1}{256}$ can be written as $\dfrac{1}{{{2}^{8}}}={{2}^{-8}}$.

Now, ${{\log }_{2}}{{2}^{-8}}=(-8\times {{\log }_{2}}2)=(-8\times 1)=-8$.

Also, $0.3=\dfrac{3}{10}$. Therefore, ${{\log }_{9}}0.3={{\log }_{9}}\left( \dfrac{3}{10} \right)={{\log }_{{{3}^{2}}}}\left( \dfrac{3}{10} \right)=\dfrac{1}{2}\left( {{\log }_{3}}3-{{\log }_{3}}10 \right)=\dfrac{1}{2}\left( 1-{{\log }_{3}}10 \right)$.

Note: we would not have easily solved the question if we would not have converted $\dfrac{1}{256}$ into exponent form. Log 10 to the base 3 can be calculated by using a calculator. Fractional conversion of 0.3 was necessary.

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