We start with the equation:
\[ \log k - \log(k-2) = \log 5 \]
Using the property of logarithms that states \(\log a - \log b = \log\left(\frac{a}{b}\right)\), we can rewrite the equation as:
\[ \log\left(\frac{k}{k-2}\right) = \log 5 \]
Since the logarithms are equal, we can set the arguments equal to each other:
\[ \frac{k}{k-2} = 5 \]
To eliminate the fraction, we cross-multiply:
\[ k = 5(k - 2) \]
Expanding the right side:
\[ k = 5k - 10 \]
Next, we bring all terms involving \(k\) to one side:
\[ k - 5k = -10 \]
This simplifies to:
\[ -4k = -10 \]
Dividing both sides by \(-4\):
\[ k = \frac{10}{4} = \frac{5}{2} \]
Thus, the value of \(k\) is:
\[ \boxed{\frac{5}{2}} \]
To verify, we can substitute \(k = \frac{5}{2}\) back into the original equation:
Calculate \(k - 2\):
\[ k - 2 = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2} \]
Now, substitute into the logarithmic expression:
\[ \log\left(\frac{5/2}{1/2}\right) = \log(5) \]
Since \(\frac{5/2}{1/2} = 5\), we have:
\[ \log 5 = \log 5 \]
This confirms that our solution is correct:
\[ \boxed{\frac{5}{2}} \]
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