To solve the equation \(\log(2x^3) + \log\left(\frac{6}{x}\right) = 6.278\), we can follow these steps:

### Step 1: Apply Logarithm Properties
Use the logarithm properties to simplify the expression:

1. **Product Property**: \(\log(a) + \log(b) = \log(ab)\)
2. **Quotient Property**: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)

The equation becomes:
\[
\log\left( (2x^3) \times \frac{6}{x} \right) = 6.278
\]

### Step 2: Simplify the Argument
Simplify the expression inside the logarithm:
\[
(2x^3) \times \frac{6}{x} = 12x^2
\]

Now the equation is:
\[
\log(12x^2) = 6.278
\]

### Step 3: Remove the Logarithm
To remove the logarithm, rewrite the equation in its exponential form:
\[
12x^2 = 10^{6.278}
\]

### Step 4: Solve for \(x\)
Calculate \(10^{6.278}\) and then solve for \(x\):

\[
x^2 = \frac{10^{6.278}}{12}
\]

\[
x = \pm \sqrt{\frac{10^{6.278}}{12}}
\]

### Step 5: Numerical Calculation
Now, let's calculate the value:

\[
10^{6.278} \approx 1901747.41
\]
\[
x^2 = \frac{1901747.41}{12} \approx 158478.951
\]
\[
x = \pm \sqrt{158478.951} \approx \pm 398.095
\]

### Final Answer:
\[
x \approx 398.095 \quad \text{or} \quad x \approx -398.095
\]

Would you like more details on any of these steps or have any questions?

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Here are 5 related questions for practice:

1. Solve the equation \(\log(x^2) + \log(x) = 5\).
2. If \(\log(3x) + \log(2/x) = 4\), what is the value of \(x\)?
3. What is the value of \(x\) if \(\log(5x^4) = 8.1\)?
4. Solve for \(x\) in the equation \(\log(7x^2) - \log(x) = 3.5\).
5. If \(\log(4x) + \log(3/x^2) = 2\), find the value of \(x\).

**Tip:** When solving logarithmic equations, remember to always check that the solutions make the arguments of the logarithms positive, as logarithms are only defined for positive real numbers.

Learn how to solve the logarithmic equation log(2x^3) + log(6/x) = 6.278 step-by-step with detailed explanations. This problem involves logarithm properties and exponential equations. Suitable for advanced high school students.

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