The problem asks us to solve the logarithmic equation $8 + \log(16x) = 36 - 3\log(x)$ for $x$.

1. Problem Description

The problem asks us to solve the logarithmic equation $8 + \log(16x) = 36 - 3\log(x)$ for $x$.

2. Solution Steps

First, let's isolate the logarithmic terms:

$8 + \log(16x) = 36 - 3\log(x)$

$\log(16x) + 3\log(x) = 36 - 8$

$\log(16x) + 3\log(x) = 28$

Now, use the logarithm properties to simplify the equation. Recall that $n\log(a) = \log(a^n)$ and $\log(a) + \log(b) = \log(ab)$. Thus,

$\log(16x) + \log(x^3) = 28$

$\log(16x \cdot x^3) = 28$

$\log(16x^4) = 28$

Assuming the base of the logarithm is 10, we can rewrite the equation in exponential form:

$16x^4 = 10^{28}$

$x^4 = \frac{10^{28}}{16}$

$x^4 = \frac{10^{28}}{2^4}$

$x^4 = (\frac{10^7}{2})^4$

Taking the fourth root of both sides:

$x = \frac{10^7}{2}$

$x = \frac{10,000,000}{2}$

$x = 5,000,000$

3. Final Answer

$5000000$