Solve each equation. -10cdot 6^7z=-5 (log_(6)frac (1)/(2))(7) log_(5)17-6 -(log_(7)frac (6)/(5))(10) log_(7)(11)/(10)

Let's solve each equation step by step.<br /><br />1. $-10 \cdot 6^{7z} = -5$<br /><br />Divide both sides by -10:<br />$6^{7z} = \frac{1}{2}$<br /><br />Take the logarithm of both sides with base 6:<br />$7z = \log_{6}\frac{1}{2}$<br /><br />Divide both sides by 7:<br />$z = \frac{\log_{6}\frac{1}{2}}{7}$<br /><br />2. $\frac{\log_{6}\frac{1}{2}}{7}$<br /><br />This expression is already simplified, so the answer is $\frac{\log_{6}\frac{1}{2}}{7}$.<br /><br />3. $\log_{5}17 - 6$<br /><br />This expression is already simplified, so the answer is $\log_{5}17 - 6$.<br /><br />4. $-\frac{\log_{7}\frac{6}{5}}{10}$<br /><br />This expression is already simplified, so the answer is $-\frac{\log_{7}\frac{6}{5}}{10}$.<br /><br />5. $\log_{7}\frac{11}{10}$<br /><br />This expression is already simplified, so the answer is $\log_{7}\frac{11}{10}$.