\(\ln \frac{\beta}{\alpha}\)

Calculus Level 2

For two real numbers \(a\) and \(b\) (\(a < b\)) in the closed interval \([0,1],\) if \[\frac{e^{7b}-e^{7a}}{b-a}=k,\] the range of the constant \(k\) can be expressed as \(\alpha < k < \beta.\) What is the value of \(\ln \frac{\beta}{\alpha}?\)

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