To solve the equation $\log(x) - \log(x - 1) = \log(4)$, we can use the properties of logarithms.

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule for logarithms states that:

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

So, applying this to the left-hand side of the equation:

$\log\left(\frac{x}{x - 1}\right) = \log(4)$

Step 2: Set the Arguments Equal

Since the logarithms of two expressions are equal, the expressions themselves must be equal:

$\frac{x}{x - 1} = 4$

Step 3: Solve the Equation

Now, solve for $x$:

1. Multiply both sides by $(x - 1)$ to get rid of the fraction:

$x = 4(x - 1)$

2. Distribute on the right-hand side:

$x = 4x - 4$

3. Move all the terms involving $x$ to one side:

$x - 4x = -4$

$-3x = -4$

4. Divide by $-3$ to isolate $x$:

$x = \frac{4}{3}$

Step 4: Check the Solution

We should check whether the solution satisfies the original logarithmic equation. The original equation is:

$\log\left(\frac{4/3}{4/3 - 1}\right) = \log(4)$

Simplifying inside the logarithm:

$\frac{4/3}{4/3 - 1} = \frac{4/3}{1/3} = 4$

So the equation holds true, confirming that the solution is correct.

Final Answer:

$x = \frac{4}{3}$

Would you like more details or have any questions about this solution? Here are some related questions:

1. How does the quotient rule for logarithms work in other logarithmic equations?
2. What are the properties of logarithms, and how are they used in solving equations?
3. Can you provide an example of a logarithmic equation involving addition?
4. How would the equation change if it were $\log(x + 1) - \log(x) = \log(4)$?
5. What if the base of the logarithms was not 10 but another number?
6. How would you solve the equation if $\log(x) + \log(x - 1) = \log(4)$?
7. What are the potential pitfalls when solving logarithmic equations?
8. How can logarithms be applied in real-life problems?

Tip: Always check if the solution to a logarithmic equation is valid by substituting it back into the original equation.