Question

If \(log_45=(log_4y)(log_6\sqrt5)\),then \(y\) equals

Answer 1

Given:
\[ \log_4 5 = (\log_4 y)(\log_6 \sqrt{5}) \]

Applying the change of base formula, \( \log_a b = \frac{\log b}{\log a} \), we get:
\[ \log_6 \sqrt{5} = \frac{\log \sqrt{5}}{\log 6} \]

Since \( \log \sqrt{5} = \frac{1}{2} \log 5 \), we have:
\[ \log_6 \sqrt{5} = \frac{1}{2} \cdot \frac{\log 5}{\log 6} \]

Substituting this into the original equation:
\[ \log_4 5 = (\log_4 y) \cdot \left(\frac{1}{2} \cdot \frac{\log 5}{\log 6}\right) \]

Using the change of base formula for the left-hand side, \( \log_4 5 = \frac{\log 5}{\log 4} \). The equation becomes:
\[ \frac{\log 5}{\log 4} = (\log_4 y) \cdot \frac{1}{2} \cdot \frac{\log 5}{\log 6} \]

Cancelling \( \log 5 \) from both sides:
\[ \frac{1}{\log 4} = \frac{1}{2 \log 6} \cdot \log_4 y \]

Multiplying both sides by \( 2 \log 6 \):
\[ \frac{2 \log 6}{\log 4} = \log_4 y \]

Expressing \( 2 \log 6 \) as \( \log 36 \):
\[ \log_4 y = \frac{\log 36}{\log 4} \]

Using the change of base formula in reverse:
\[ \log_4 y = \log_{4} 36 \Rightarrow y = 36 \]

∴ The value of \( y \) is 36.