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We are given that \( a > b > 0 \) and the equation:

\[
\log a - \log b = \log(a - b)
\]

We need to show that:

\[
a = \frac{b^2}{b - 1}
\]

### Step 1: Simplify the logarithmic equation

Using the logarithmic property \( \log x - \log y = \log \left( \frac{x}{y} \right) \), we can simplify the left-hand side:

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

### Step 2: Exponentiate both sides

Since the logarithms are equal, we can remove the logarithms by exponentiating both sides:

\[
\frac{a}{b} = a - b
\]

### Step 3: Solve for \( a \)

Multiply both sides by \( b \) to eliminate the denominator:

\[
a = b(a - b)
\]

Distribute \( b \) on the right-hand side:

\[
a = ab - b^2
\]

Now, isolate \( a \) on one side by subtracting \( ab \) from both sides:

\[
a - ab = -b^2
\]

Factor out \( a \) on the left-hand side:

\[
a(1 - b) = -b^2
\]

Finally, solve for \( a \) by dividing both sides by \( 1 - b \):

\[
a = \frac{-b^2}{1 - b}
\]

Since \( 1 - b \) is negative, this simplifies to:

\[
a = \frac{b^2}{b - 1}
\]

Thus, we have shown that:

\[
a = \frac{b^2}{b - 1}
\]

Would you like any further clarification or more details on this?

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Here are some related questions to consider:

1. How does the property of logarithms simplify this equation?
2. What happens if we don't assume \( a > b > 0 \)?
3. Can this result be applied to other forms of logarithmic equations?
4. How would this result change if we used natural logarithms instead of base-10 logarithms?
5. What are the conditions on \( b \) to ensure \( a \) remains positive?

**Tip**: When dealing with logarithmic equations, always check whether using properties like \( \log x - \log y = \log \frac{x}{y} \) simplifies the equation early on.

Given that a > b > 0 and that a and b satisfy the equation log a - log b = log(a - b), show that a = b^2 / (b - 1).

Explore a detailed solution proving that a = b^2 / (b - 1) from the logarithmic equation log a - log b = log(a - b), using logarithmic properties and algebraic manipulation. Ideal for students at an undergraduate math level.

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