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^(5)y)) 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^(5)y)) Answer:

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Identify Properties: Identify the properties used to expand log((x)/(z^(5)y)). 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 log((x)/(z^(5)y)). Using the quotient property, we can separate the logarithm of the quotient into the difference of the logarithms of the numerator and the denominator. log((x)/(z^(5)y)) = log(x) - log(z^(5)y)Apply Quotient Property Again: Apply the quotient property again to separate log(z^(5)y). We can further expand log(z^(5)y) into log(z^(5)) + log(y) using the product property of logarithms, which states that the logarithm of a product is the sum of the logarithms. log(z^(5)y) = log(z^(5)) + log(y)Apply Power Property: Apply the power property to log(z^(5)). Using the power property, we can take the exponent out in front of the logarithm. log(z^(5)) = 5 * log(z)Substitute Expanded Log: Substitute the expanded log(z^(5)) back into the original expression. Now we replace log(z^(5)) in our expression from Step 2 with 5 * log(z). log((x)/(z^(5)y)) = log(x) - (5 * log(z) + log(y))Distribute Negative Sign: Distribute the negative sign to both terms in the parentheses. We need to apply the negative sign to both log(z^(5)) and log(y). log((x)/(z^(5)y)) = log(x) - 5 * log(z) - log(y)

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

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