Calculate ln(4) with Mixed Base-2 and Base-7 Logarithmic Expression

Calculate the value of the following expression:

$\ln4\times(\log_7x^7-\log_7x^4-\log_7x^3+\log_2y^4-\log_2y^3-\log_2y)$

To solve this problem, we'll follow these steps:

• Simplify the logarithmic expression using logarithmic identities.
• Substitute the simplified result back into the main expression and calculate its value.

Now, let's work through each step:

Step 1: Simplify the logarithmic expression. We'll simplify the parts involving $\log_7$ first, then those involving $\log_2$.

For the terms with $\log_7$:
- Convert $\log_7 x^n$ terms using the power rule: $\log_7 x^7 = 7 \log_7 x$, $\log_7 x^4 = 4 \log_7 x$, and $\log_7 x^3 = 3 \log_7 x$.
- The expression becomes $7 \log_7 x - 4 \log_7 x - 3 \log_7 x$.
- Simple arithmetic yields $0 \log_7 x$, which simplifies to $0$.

For the terms with $\log_2$:
- Similarly, $\log_2 y^n$ terms use the power rule: $\log_2 y^4 = 4 \log_2 y$, $\log_2 y^3 = 3 \log_2 y$, and $\log_2 y = 1 \log_2 y$.
- The expression is $4 \log_2 y - 3 \log_2 y - 1 \log_2 y$.
- Simple arithmetic gives $0 \log_2 y$, which also simplifies to $0$.

Step 2: Substitute these back into the original expression:

Original expression:
$\ln 4 \times (0 + 0) = \ln 4 \times 0 = 0$.

Therefore, the value of the expression is $\textbf{0}$.