Question 21

Write as the sum and/or difference of logarithms. Express powers as factors. $\log _{8}\left(x z^{3}\right)$
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Step 1 ：The given expression is a logarithm with base 8 and argument x*z^3.

Step 2 ：According to the properties of logarithms, we can express this as the sum of the logarithms of x and z^3.

Step 3 ：Furthermore, we can express the power of z as a factor.

Step 4 ：So, we need to convert \(\log _{8}\left(x z^{3}\right)\) into a sum of logarithms.

Step 5 ：This can be done as follows: \(\log(x*z**3)/\log(8)\) is equivalent to \(\log(x)/\log(8) + \log(z**3)/\log(8)\).

Step 6 ：Finally, we can express the power of z as a factor, giving us \(\log(x)/\log(8) + 3*\log(z)/\log(8)\).

Step 7 ：Final Answer: \(\boxed{\log _{8} x+3 \log _{8} z}\)