Solve Complex Logarithm Equation: log₂ₐ(e⁷) with Natural Logarithms

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

• Step 1: Simplify the left side of the equation.
• Step 2: Simplify the right side of the equation.
• Step 3: Set the two sides equal and solve for $X$.

Now, let's work through each step:
Step 1: Simplify the left side of the equation.
Given: $\log_{2a}(e^7(\ln a+\ln 4a))$.
Combine the logarithms: $\ln 4a = \ln 4 + \ln a$.
Thus, $\ln a + \ln 4a = \ln a + \ln 4 + \ln a = 2\ln a + \ln 4$.
So, $e^7(2\ln a + \ln 4) = e^{7}e^{2\ln a}e^{\ln 4}$.
This simplifies to $e^{7}a^2 \cdot 4$.
Therefore, the left side is: $\log_{2a}(4a^2e^7)$.

Step 2: Simplify the right side of the equation.
Given: $\log_4 x - \log_4 x^2 + \log_4 \frac{1}{x+1}$.
Combining using the quotient and power rules: $\log_4 \frac{x}{x^2} + \log_4 \frac{1}{x+1}$.
Further simplify: $\log_4 \frac{1}{x(x+1)}$.

Step 3: Set the two sides equal and solve for $X$.
We have: $\log_{2a}(4a^2e^7) = \log_4 \frac{1}{x(x+1)}$.
Rewriting with change of base: $\frac{\ln(4a^2e^7)}{\ln(2a)} = -\log_4(x(x+1))$.
Substitute known values and solve: $4a^2e^7 = 1/(x^2+x)$.
Framing: Solve $x^2 + x - (4a^2e^7) = 0$.

The solution for $X$ is found by applying the quadratic formula:

Therefore, the solution to the problem is $X = -\frac{1}{2}+\frac{\sqrt{1+4^{-13}}}{2}$.