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?

---

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.