To solve the equation $2 \log(x) - \log(7x - 1) = 0$, we can follow these steps:

Step 1: Combine the logarithms

First, we'll use the logarithm property:

$a \log(b) = \log(b^a)$

So, the equation becomes:

$\log(x^2) - \log(7x - 1) = 0$

Step 2: Apply the quotient rule of logarithms

Using the property:

$\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$

We can rewrite the equation as:

$\log\left(\frac{x^2}{7x - 1}\right) = 0$

Step 3: Exponentiate both sides to eliminate the logarithm

Since $\log(y) = 0$ implies $y = 1$:

$\frac{x^2}{7x - 1} = 1$

Step 4: Solve the resulting equation

Multiply both sides by $7x - 1$ to clear the fraction:

$x^2 = 7x - 1$

Rearrange the equation to set it to zero:

$x^2 - 7x + 1 = 0$

Step 5: Solve the quadratic equation

We can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = -7$, and $c = 1$. Substituting these values:

$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(1)}}{2(1)}$

$x = \frac{7 \pm \sqrt{49 - 4}}{2}$

$x = \frac{7 \pm \sqrt{45}}{2}$

$x = \frac{7 \pm 3\sqrt{5}}{2}$

So, the solutions are:

$x = \frac{7 + 3\sqrt{5}}{2} \quad \text{or} \quad x = \frac{7 - 3\sqrt{5}}{2}$

Step 6: Check for extraneous solutions

Since $x$ must satisfy the original logarithmic conditions (i.e., the arguments of the logarithms must be positive), we need to check:

For $x = \frac{7 + 3\sqrt{5}}{2}$, $x > 0$ and $7x - 1 > 0$, so this is a valid solution.

For $x = \frac{7 - 3\sqrt{5}}{2}$, $x$ is positive but smaller than 1, making $7x - 1$ negative, which is not valid.

Thus, the only valid solution is:

$\boxed{x = \frac{7 + 3\sqrt{5}}{2}}$

Would you like more details or have any questions?

Related Questions:

1. How do logarithmic properties help simplify equations?
2. What is the discriminant in the quadratic formula, and what does it indicate?
3. What are the general conditions for logarithmic arguments to be valid?
4. Can a quadratic equation have no real solutions? Why or why not?
5. How does the quadratic formula derive from completing the square?

Tip: Always check the domain restrictions when solving logarithmic equations to ensure that all solutions are valid.