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To solve the equation \( \ln 7 + \ln -5x = \ln 77 \), we start by combining the logarithms on the left using the product rule of logarithms: \[ \ln(7 \cdot -5x) = \ln 77. \] This gives us: \[ \ln(-35x) = \ln 77. \] Now, since the logarithmic function is one-to-one, we can set the arguments equal to each other: \[ -35x = 77. \] Next, solve for \( x \): \[ x = \frac{-77}{35} = -\frac{11}{5}. \] Remember that the logarithm is only defined for positive arguments, so we must also consider whether \(-5x\) is positive. Since \(x = -\frac{11}{5}\), we have: \[ -5x = -5 \left(-\frac{11}{5}\right) = 11, \] which is positive. Thus, \(x = -\frac{11}{5}\) is indeed a valid solution for the original equation. So the solution is: \[ x = -\frac{11}{5}. \]
