Problem
Write the expression as a single logarithm. \[ 5 \log _{c} w-\frac{1}{3} \log _{c} z+7 \log _{c} x \]
Solution
Step 1 ：First, we apply the power rule of logarithms, which states that \(\log_b m^n = n \log_b m\). This gives us \(\log_c w^5 - \log_c z^{1/3} + \log_c x^7\).
Step 2 ：Next, we apply the product rule of logarithms, which states that \(\log_b mn = \log_b m + \log_b n\). This gives us \(\log_c (w^5 x^7) - \log_c z^{1/3}\).
Step 3 ：Finally, we apply the quotient rule of logarithms, which states that \(\log_b \frac{m}{n} = \log_b m - \log_b n\). This gives us \(\log_c \frac{w^5 x^7}{z^{1/3}}\).
Step 4 ：So, the expression \(5 \log _{c} w-\frac{1}{3} \log _{c} z+7 \log _{c} x\) can be written as a single logarithm as \(\boxed{\log_c \frac{w^5 x^7}{z^{1/3}}}\).
