Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x,log y, and log z. log ((x)/(z^(4)root(3)(y^(4)))) Answer:

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Expand the logarithm fully using the properties of logs. Express the final answer in terms of  log x,log y, and  log z.  log ((x)/(z^(4)root(3)(y^(4)))) Answer:

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Identify Properties: Identify the properties used to expand the logarithm. We will use the quotient property of logarithms to separate the numerator and the denominator, and the power property to bring down the exponents. The properties are as follows: Quotient Property: log_b (P/Q) = log_b P - log_b Q Power Property: log_b (P^k) = k * log_b PApply Quotient Property: Apply the quotient property to the given logarithm. Using the quotient property, we can write log((x)/(z^(4)root(3)(y^(4)))) as log(x) - log(z^(4)root(3)(y^(4))).Apply Power Property: Apply the power property to the logarithm of the denominator. We have log(z^(4)root(3)(y^(4))). Since z is raised to the power of 4, we can bring the exponent down in front of the log, which gives us 4 * log(z). However, we need to be careful with the cube root of y^4, which is y^(4/3). We can also bring this exponent down in front of the log, which gives us (4/3) * log(y).Combine Results: Combine the results from Step 2 and Step 3. We now have log(x) - (4 * log(z) + (4/3) * log(y)). We need to distribute the negative sign to both terms in the parentheses.Distribute and Simplify: Distribute the negative sign and simplify. Distributing the negative sign gives us log(x) - 4 * log(z) - (4/3) * log(y).

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{x}{z^{4} \sqrt[3]{y^{4}}}$ $\newline$ Answer:

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