To solve the given equation
[tex]\[
\log (20x^3) - 2 \log (x) = 4,
\][/tex]
we will use the properties of logarithms to simplify and solve for [tex]\(x\)[/tex].
### Step 1: Use the property of logarithms:
First, recall the logarithm property:
[tex]\[
\log (a \cdot b) = \log a + \log b.
\][/tex]
Applying this property to [tex]\(\log (20x^3)\)[/tex], we get:
[tex]\[
\log (20x^3) = \log 20 + \log x^3.
\][/tex]
### Step 2: Simplify [tex]\(\log x^3\)[/tex]:
Using another property of logarithms,
[tex]\[
\log x^3 = 3\log x,
\][/tex]
we rewrite the equation as:
[tex]\[
\log 20 + 3\log x - 2\log x = 4.
\][/tex]
### Step 3: Combine like terms:
Combine the logarithmic terms involving [tex]\(x\)[/tex]:
[tex]\[
\log 20 + 3\log x - 2 \log x = \log 20 + \log x = 4.
\][/tex]
### Step 4: Use the property of logarithms again:
Using the property that [tex]\(\log a + \log b = \log (ab)\)[/tex], we have:
[tex]\[
\log (20x) = 4.
\][/tex]
### Step 5: Convert from logarithmic form to exponential form:
Recall that [tex]\(\log a = b \implies 10^b = a\)[/tex]. So,
[tex]\[
20x = 10^4.
\][/tex]
Evaluate [tex]\(10^4\)[/tex]:
[tex]\[
10^4 = 10000.
\][/tex]
Thus,
[tex]\[
20x = 10000.
\][/tex]
### Step 6: Solve for [tex]\(x\)[/tex]:
Divide both sides by 20 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{10000}{20} = 500.
\][/tex]
### Conclusion:
The value of [tex]\(x\)[/tex] that satisfies the equation
[tex]\[
\log (20x^3) - 2 \log (x) = 4
\][/tex]
is [tex]\(x = 500\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{500}.
\][/tex]
