Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x,log y, and log z. log ((y)/(z^(5)x)) 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 ((y)/(z^(5)x)) Answer:

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Identify Properties: Identify the properties used to expand log((y)/(z^(5)x)). We will use the quotient property and the power property of logarithms to expand the given expression. 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 separate the numerator and the denominator of the fraction inside the logarithm. log((y)/(z^(5)x)) = log(y) - log(z^(5)x)Separate z^(5) and x: Apply the quotient property again to separate z^(5) and x. Now we have log(z^(5)x), which is a product inside the logarithm. We can use the quotient property again to separate them. log(z^(5)x) = log(z^(5)) + log(x)Apply Power Property: Apply the power property to the term log(z^(5)). Using the power property, we can take the exponent out in front of the log. log(z^(5)) = 5 * log(z)Substitute Expanded Term: Substitute the expanded term back into the original expression. Now we can replace log(z^(5)x) in our original expression with the expanded terms. log((y)/(z^(5)x)) = log(y) - (5 * log(z) + log(x))Distribute Negative Sign: Distribute the negative sign to both terms in the parentheses. When we distribute the negative sign, we get: log((y)/(z^(5)x)) = log(y) - 5 * log(z) - log(x)

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{y}{z^{5} x}$ $\newline$ Answer:

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