Answer:
[tex]\textsf{C)}\quad \log \left(5^6 \sqrt[3]{12}\right)[/tex]
Step-by-step explanation:
Given logarithmic expression:
[tex]6 \log 5 + \dfrac{\log 12}{3}[/tex]
Dividing the logarithm of a number by a constant is equivalent to multiplying the logarithm of that number by the reciprocal of the constant. Therefore, the original expression can be rewritten as:
[tex]6 \log 5 + \dfrac{1}{3}\log 12[/tex]
Apply the power rule of logarithms, which states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base number:
[tex]\log 5^6 + \log 12^\frac{1}{3}[/tex]
Apply the the fractional exponent rule to the argument of the second term:
[tex]\log 5^6 + \log \sqrt[3]{12}[/tex]
Finally, apply the product rule of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the factors:
[tex]\log \left(5^6 \sqrt[3]{12}\right)[/tex]
Therefore, the given expression condensed into a single logarithm is:
[tex]\Large\boxed{\boxed{\log \left(5^6 \sqrt[3]{12}\right)}}[/tex]
[tex]\dotfill[/tex]
Rules used
[tex]\boxed{\begin{array}{c}\underline{\textsf{Power Rule of Logarithms}}\\\\\large\text{$\log x^n=n\log x$}\end{array}}[/tex]
[tex]\boxed{\begin{array}{c}\underline{\textsf{Fractional Exponent Rule}}\\\\\Large\text{$a^{\frac{m}{n}}=\sqrt[n]{a^m}$}\end{array}}[/tex]
[tex]\boxed{\begin{array}{c}\underline{\textsf{Product Rule of Logarithms}}\\\\\large\text{$\log xy=\log x + \log y$}\end{array}}[/tex]
