Question

$\log_{6} 216 \sqrt{6}$ is:

• A. $3$
• B. $\frac{3}{2}$
• C. $\frac{7}{2}$
• D. None of these

Hint:

Always try to express numbers as powers of the base when solving logarithmic problems. This simplifies the calculation.

Answer 1

The Correct Option is C

We are asked to evaluate $\log_{6} 216 \sqrt{6}$.
First, we note that $216 = 6^{3}$. This is because $6^{3} = 6 \times 6 \times 6 = 216$.
Also, $\sqrt{6} = 6^{1/2}$.
Therefore:
\[ 216 \sqrt{6} = 6^{3} \times 6^{1/2} = 6^{3 + 1/2} = 6^{7/2} \] Now:
\[ \log_{6} (6^{7/2}) = \frac{7}{2} \log_{6} 6 \] Since $\log_{6} 6 = 1$, we have:
\[ \log_{6} (6^{7/2}) = \frac{7}{2} \] Hence, the value is $\frac{7}{2}$.