Problem

Write the expression as a single logarithm. \[ \frac{1}{5} \log _{8} z+6 \log _{8} x-\log _{8} y \]

Solution

Step 1 :Given the expression \(\frac{1}{5} \log _{8} z+6 \log _{8} x-\log _{8} y\)

Step 2 :We can use the properties of logarithms to simplify this expression. The properties are as follows:

Step 3 :1. \(a \log_b c = \log_b (c^a)\)

Step 4 :2. \(\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)\)

Step 5 :3. \(\log_b a + \log_b c = \log_b (ac)\)

Step 6 :Applying these properties, the given expression can be simplified as follows:

Step 7 :\(\frac{1}{5} \log _{8} z+6 \log _{8} x-\log _{8} y = \log _{8} (z^{1/5}) + \log _{8} (x^6) - \log _{8} y\)

Step 8 :This can be further simplified using the properties 2 and 3 to:

Step 9 :\(\log _{8} \left(\frac{z^{1/5} x^6}{y}\right)\)

Step 10 :Final Answer: The expression can be written as a single logarithm as follows:

Step 11 :\(\boxed{\log _{8} \left(\frac{z^{1/5} x^6}{y}\right)}\)

From Solvely APP
Source: https://solvelyapp.com/problems/7726/

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