log3 8+log3 18/ 2log3 4+log3 2

To simplify the expression \(\log_3 8 + \frac{\log_3 18}{2 \log_3 4 + \log_3 2}\), we can start by breaking it down.

First, recall that:

• \(\log_3 8 = \log_3 (2^3) = 3 \log_3 2\)
• \(\log_3 18 = \log_3 (2 \cdot 3^2) = \log_3 2 + 2 \log_3 3\)
• \(\log_3 4 = \log_3 (2^2) = 2 \log_3 2\)

Now substitute these back into the expression:

1. Calculate \(2 \log_3 4 + \log_3 2\): \[ 2 \log_3 4 = 2(2 \log_3 2) = 4 \log_3 2 \] Thus, \[ 2 \log_3 4 + \log_3 2 = 4 \log_3 2 + \log_3 2 = 5 \log_3 2 \]

2. Now we can rewrite the entire expression: \[ \log_3 8 + \frac{\log_3 18}{2 \log_3 4 + \log_3 2} = 3 \log_3 2 + \frac{\log_3 2 + 2 \log_3 3}{5 \log_3 2} \]

Breaking it down further: \[ = 3 \log_3 2 + \frac{\log_3 2}{5 \log_3 2} + \frac{2 \log_3 3}{5 \log_3 2} \] \[ = 3 \log_3 2 + \frac{1}{5} + \frac{2 \log_3 3}{5 \log_3 2} \] Now, let’s find a common denominator for all terms: \[ = \frac{15 \log_3 2}{5} + \frac{1}{5} + \frac{2 \log_3 3}{5 \log_3 2} \] The three fractions can be combined: \[ = \frac{15 \log_3 2 + 1 + 2 \frac{\log_3 3}{\log_3 2}}{5} \]

This expression represents the final simplified result but can vary based on whether additional numeric evaluation is needed or specific log values are pursued. If specific numerical values for \(\log_3 2\) and \(\log_3 3\) are provided, the evaluation can yield a definitive numeric result.

Do you want a numeric approximation or further simplification?