Write the expression as a single logarithm. \[ 7 \log _{c}(7 x+1)+\frac{1}{2} \log _{c}(x+8) \]

Step 1 ：Given the expression \(7 \log _{c}(7 x+1)+\frac{1}{2} \log _{c}(x+8)\)

Step 2 ：We can use the properties of logarithms to simplify this expression. The properties are as follows: \(a \log_b(c) = \log_b(c^a)\) and \(\log_b(c) + \log_b(d) = \log_b(cd)\)

Step 3 ：Applying the first property to both terms, we get \(\log_c((7x+1)^7) + \log_c((x+8)^{1/2})\)

Step 4 ：Then, applying the second property to combine the two terms into a single logarithm, we get \(\log_c((7x+1)^7 * (x+8)^{1/2})\)

Step 5 ：The simplified expression is \(\boxed{\log _{c}\left((7 x+1)^{7}(x+8)^{\frac{1}{2}}\right)}\)