Solve: 2log₃4 × log₂9 Logarithm Product Problem

$2\log_34\times\log_29=$

To solve this problem, we need to evaluate $2\log_3 4 \times \log_2 9$. We'll use the change of base formula to simplify the logarithms.

• Step 1: Apply the change of base formula to both logarithms.
• Step 2: Simplify the expressions by substituting appropriate values.
• Step 3: Compute the multiplication of the simplified values.

Step 1: Convert the logarithms using the change of base formula:

$\log_3 4 = \frac{\log_{10} 4}{\log_{10} 3}$ and $\log_2 9 = \frac{\log_{10} 9}{\log_{10} 2}$.

Step 2: Substitute these back into the expression:

$2 \times \frac{\log_{10} 4}{\log_{10} 3} \times \frac{\log_{10} 9}{\log_{10} 2}$.

Recognize that $\log_{10} 4 = 2 \log_{10} 2$ and $\log_{10} 9 = 2 \log_{10} 3$, hence simplifying gives:

= $2 \times \frac{2 \log_{10} 2}{\log_{10} 3} \times \frac{2 \log_{10} 3}{\log_{10} 2}$.

Step 3: Cancel terms and calculate:

The terms $\log_{10} 2$ and $\log_{10} 3$ cancel out:

= $2 \times 2 \times 2 = 8$.

Therefore, the solution to the problem is $\boxed{8}$, which corresponds to choice 3 in the provided answer choices.