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

1. Problem Description

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

2. Solution Steps

We are given the equation $8 + \log(16x) = 36 - 3\log(x)$.

Subtracting 8 from both sides gives

$\log(16x) = 28 - 3\log(x)$.

We can use the property $\log(ab) = \log(a) + \log(b)$ to rewrite $\log(16x)$ as $\log(16) + \log(x)$.

So, we have $\log(16) + \log(x) = 28 - 3\log(x)$.

Adding $3\log(x)$ to both sides yields

$\log(16) + 4\log(x) = 28$.

Subtracting $\log(16)$ from both sides, we have

$4\log(x) = 28 - \log(16)$.

Dividing by 4 gives

$\log(x) = 7 - \frac{1}{4}\log(16)$.

Using the power rule of logarithms, $\log(a^b) = b\log(a)$, we can rewrite the equation as:

$\log(x) = 7 - \log(16^{1/4})$.

Since $16^{1/4} = \sqrt[4]{16} = 2$, we have

$\log(x) = 7 - \log(2)$.

Then $\log(x) = \log(10^7) - \log(2)$, assuming base 10 logarithm.

Using the property $\log(a) - \log(b) = \log(\frac{a}{b})$, we get

$\log(x) = \log(\frac{10^7}{2}) = \log(\frac{10000000}{2}) = \log(5000000)$.

Therefore, $x = 5000000$.

3. Final Answer

$x = 5000000$