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Answer :
We start with the logarithm
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3}.
$$[/tex]
Step 1. Write the logarithm of a quotient as the difference of logarithms:
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3} = \log_6 (7^9 \sqrt[5]{x}) - \log_6 (y^3).
$$[/tex]
Step 2. Write the logarithm of a product as the sum of logarithms:
[tex]$$
\log_6 (7^9 \sqrt[5]{x}) = \log_6 (7^9) + \log_6 (\sqrt[5]{x}).
$$[/tex]
Thus, the expression becomes:
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3} = \log_6 (7^9) + \log_6 (\sqrt[5]{x}) - \log_6 (y^3).
$$[/tex]
Step 3. Apply the power rule for logarithms, which states that [tex]$\log_b (a^c) = c \, \log_b (a)$[/tex].
For each term:
1. [tex]$$\log_6 (7^9) = 9 \, \log_6 7.$$[/tex]
2. Since [tex]$\sqrt[5]{x} = x^{\frac{1}{5}}$[/tex], we have
[tex]$$
\log_6 (\sqrt[5]{x}) = \frac{1}{5} \, \log_6 x.
$$[/tex]
3. [tex]$$\log_6 (y^3) = 3 \, \log_6 y.$$[/tex]
Substitute these results back into the expression:
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3} = 9 \, \log_6 7 + \frac{1}{5} \, \log_6 x - 3 \, \log_6 y.
$$[/tex]
This shows that the original logarithmic expression can be written as
[tex]$$
9 \, \log_6 7 + \frac{1}{5} \, \log_6 x - 3 \, \log_6 y.
$$[/tex]
Thus, the correct answer corresponds to option A.
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3}.
$$[/tex]
Step 1. Write the logarithm of a quotient as the difference of logarithms:
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3} = \log_6 (7^9 \sqrt[5]{x}) - \log_6 (y^3).
$$[/tex]
Step 2. Write the logarithm of a product as the sum of logarithms:
[tex]$$
\log_6 (7^9 \sqrt[5]{x}) = \log_6 (7^9) + \log_6 (\sqrt[5]{x}).
$$[/tex]
Thus, the expression becomes:
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3} = \log_6 (7^9) + \log_6 (\sqrt[5]{x}) - \log_6 (y^3).
$$[/tex]
Step 3. Apply the power rule for logarithms, which states that [tex]$\log_b (a^c) = c \, \log_b (a)$[/tex].
For each term:
1. [tex]$$\log_6 (7^9) = 9 \, \log_6 7.$$[/tex]
2. Since [tex]$\sqrt[5]{x} = x^{\frac{1}{5}}$[/tex], we have
[tex]$$
\log_6 (\sqrt[5]{x}) = \frac{1}{5} \, \log_6 x.
$$[/tex]
3. [tex]$$\log_6 (y^3) = 3 \, \log_6 y.$$[/tex]
Substitute these results back into the expression:
[tex]$$
\log_6 \frac{7^9 \sqrt[5]{x}}{y^3} = 9 \, \log_6 7 + \frac{1}{5} \, \log_6 x - 3 \, \log_6 y.
$$[/tex]
This shows that the original logarithmic expression can be written as
[tex]$$
9 \, \log_6 7 + \frac{1}{5} \, \log_6 x - 3 \, \log_6 y.
$$[/tex]
Thus, the correct answer corresponds to option A.
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