91Ó°ÊÓ

By definition, the square root of a number can be represented as a power of one-half. So, the given expression becomes: \[\frac{\log_b 27^{\frac{1}{2}}}{3}\] Now, applying the power rule of logarithm which states that \(log_c (a^n) = n \cdot log_c (a)\), we can write: \[ \frac{1}{2} \log_b 27 . \frac{1}{3}\]

First, we will simplify the term inside the logarithm: \[\sqrt{27} = 27^\frac{1}{2} = 3^3^\frac{1}{2} = 3^\frac{3}{2}\] Now, the given expression becomes: \[\log_b \frac{3^\frac{3}{2}}{3}\] Applying the quotient rule of logarithm which states that \(log_c \frac{a}{b} = log_c (a) - log_c (b)\), we can write: \[\log_b 3^\frac{3}{2} - \log_b 3\]

Now we have the two simplified expressions: \[\frac{1}{2} \log_b 27 . \frac{1}{3}\] and \[\log_b 3^\frac{3}{2} - \log_b 3\] We can notice that they are equal if we can show that \(27^{\frac{1}{2}}\cdot\frac{1}{3} = 3^\frac{3}{2} - 1\). To prove this, rewrite the left side of the equation: \[\frac{1}{2} \log_b 27 . \frac{1}{3} = \frac{1}{2} \cdot (\log_b 27 - \log_b 3)\] Now, rewrite the right side of the equation using the power rule of logarithm: \[\log_b 3^\frac{3}{2} - \log_b 3 = \frac{3}{2} \log_b 3 - \log_b 3\] Now both the expressions are equal, as we can see that: \[\frac{1}{2} . (\log_b 27 - \log_b 3) = \frac{3}{2} \log_b 3 - \log_b 3\] Thus, we have shown that \(\frac{\log_b \sqrt{27}}{3} = \log_b \frac{\sqrt{27}}{3}\) for every positive number \(b \neq 1\).