Solution

Write the following as a single logarithm

$$\frac{1}{2} \log_5 49 - \frac{1}{3} \log_5 8 + 13 \log_5 1$$


$$\begin{array}{rcll} \frac{1}{2} \log_5 49 - \frac{1}{3} \log_5 8 + 13 \log_5 1 &=& \frac{1}{2} \log_5 49 - \frac{1}{3} \log_5 8 & \scriptsize{\textrm{recall } 5^0 = 1, \textrm{ so } \log_5 1 = 0}\\\\ &=& \log_5 49^{1/2} - \log_5 8^{1/3} & \scriptsize{\textrm{since } \log_b x^n = n\log_b x}\\\\ &=& \log_5 7 - \log_5 2\\\\ &=& \log_5 \left( \frac{7}{2} \right) & \scriptsize{\textrm{since } \log_b \frac{x}{y} = \log_b x - \log_b y} \end{array}$$