Solve the Complex Logarithmic Equation: (log_x4 + log_x30.25)/(x*log_x11) + x = 3

$\frac{\log_x4+\log_x30.25}{x\log_x11}+x=3$

$x=\text{?}$

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

• Step 1: Simplify the logarithmic expression $\log_x4 + \log_x30.25$.
• Step 2: Use the change of base formula for logarithms.
• Step 3: Substitute and solve for $x$.

Now, let's work through each step:
Step 1: Simplify the logarithmic expression by using the property $\log_x4 + \log_x30.25 = \log_x(4 \times 30.25)$.
Step 2: Calculate $4 \times 30.25 = 121$, then express as $\log_x121$.

Step 3: The equation becomes $\frac{\log_x121}{x\log_x11} + x = 3$. We know $\log_x121 = 2$ when $x = 11$, thus evaluate the expression with possible values.

Consider a simpler value for $x$, like 2. calc $\log_2 4 = 2$ and $\log_2 121$. Using the logarithmic laws further simplifies if appropriate, achieving solution $x = 2$.

Therefore, the solution to the problem is $x = 2$.