Question

Simplify: $\frac{log_3 512}{log_3 8} \times \frac{log_3 3}{log_3 9} | \frac{log_3 3}{log_3 4} + \frac{log_3 3}{log_3 4}$

Answer 1

$\frac{3 \log_3 4}{4}$

Answer 2

The problem asks to simplify the given logarithmic expression. The expression is: $\frac{\log_3 512}{\log_3 8} \times \frac{\log_3 3}{\log_3 9} \left| \frac{\log_3 3}{\log_3 4} + \frac{\log_3 3}{\log_3 4} \right.$ The symbol ' | ' is unconventional in this context. Assuming it represents division, the expression can be written as: $\frac{\log_3 512}{\log_3 8} \times \frac{\log_3 3}{\log_3 9} \div \left( \frac{\log_3 3}{\log_3 4} + \frac{\log_3 3}{\log_3 4} \right)$ We will simplify each part of the expression using logarithm properties:

1. Change of Base Formula: $\frac{\log_b a}{\log_b c} = \log_c a$
2. Power Rule: $\log_b (a^k) = k \log_b a$
3. Identity: $\log_b b = 1$

Step 1: Simplify the first fraction $\frac{\log_3 512}{\log_3 8}$ Using the change of base formula, this is equivalent to $\log_8 512$. Since $8^3 = 512$, we have $\log_8 512 = 3$.

Step 2: Simplify the second fraction $\frac{\log_3 3}{\log_3 9}$ Using the identity $\log_3 3 = 1$ and the power rule for $\log_3 9$: $\log_3 9 = \log_3 (3^2) = 2 \log_3 3 = 2 \times 1 = 2$. So, the fraction becomes $\frac{1}{2}$.

Step 3: Simplify the terms inside the parenthesis $\frac{\log_3 3}{\log_3 4} + \frac{\log_3 3}{\log_3 4}$ Since $\log_3 3 = 1$, each term is $\frac{1}{\log_3 4}$. So, the sum is $\frac{1}{\log_3 4} + \frac{1}{\log_3 4} = \frac{2}{\log_3 4}$.

Step 4: Substitute the simplified terms back into the main expression The expression now becomes: $3 \times \frac{1}{2} \div \left( \frac{2}{\log_3 4} \right)$

Step 5: Perform the multiplication and division $\frac{3}{2} \div \frac{2}{\log_3 4}$ To divide by a fraction, multiply by its reciprocal: $\frac{3}{2} \times \frac{\log_3 4}{2}$ $\frac{3 \log_3 4}{4}$

Step 6: Further simplification (optional) We can express $\log_3 4$ in terms of $\log_3 2$: $\log_3 4 = \log_3 (2^2) = 2 \log_3 2$. Substitute this back into the expression: $\frac{3 \times (2 \log_3 2)}{4} = \frac{6 \log_3 2}{4} = \frac{3}{2} \log_3 2$

Both $\frac{3 \log_3 4}{4}$ and $\frac{3}{2} \log_3 2$ are simplified forms of the expression.

The final answer is $\boxed{\frac{3 \log_3 4}{4}}$.

Explanation of the solution: The expression is simplified by first interpreting the symbol '|' as division. Then, standard logarithm properties such as the change of base formula ($\frac{\log_b a}{\log_b c} = \log_c a$), power rule ($\log_b (a^k) = k \log_b a$), and identity ($\log_b b = 1$) are applied to simplify each fractional term. The simplified terms are then combined using the arithmetic operations.