Ответ

To solve the expression given, use the properties of logarithms: 1. \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\). So the expression \[ \frac{\log_7 8}{\log_7 15 - \log_7 30} \] can be simplified as: \[ \frac{\log_7 8}{\log_7\left(\frac{15}{30}\right)} \] \[ = \frac{\log_7 8}{\log_7\left(\frac{1}{2}\right)} \] Now, use the property \(\log_b(a^n) = n \log_b(a)\): \[ = \frac{\log_7 8}{-1 \cdot \log_7 2} \] \[ = -\frac{\log_7 8}{\log_7 2} \] Finally, use the property \(\log_b(m^n) = n \log_b(m)\): \[ = -\log_7(2^3) \] \[ = -3 \log_7 2 \] \[ = -3 \]