Alg9.4.6

Find the value of $\log_6(216\sqrt{6}),$

Expressing the log number in terms of the base,we get

$\log_6(6^3\sqrt{6}),$

Further simplifying

$\log_6(6^{3+\frac{1}{2}}),$

$\log_6(6^{\frac{7}{2}}),$

As per the logarithmic rule that $\log_a(a^m)=m\log_a(a)\,$

we get

$\frac{7}{2}\log_6(6),$

As per the rule that the logarithm of a number with respect to the same base number is 1

The value of given logarithm is

$\log_6(216\sqrt{6})=\frac{7}{2}\,$