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^(5))/(y^(2)z^(3))) 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^(5))/(y^(2)z^(3))) Answer:

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Identify Properties: Identify the properties used to expand log((x^(5))/(y^(2)z^(3))). We will use the quotient property of logarithms to separate the numerator and denominator, and the power property to bring down the exponents. Quotient Property: log_b(P/Q) = log_b(P) - log_b(Q) Power Property: log_b(P^k) = k * log_b(P)Apply Quotient Property: Apply the quotient property to the given logarithm. Using the quotient property, we can write log((x^(5))/(y^(2)z^(3))) as log(x^(5)) - log(y^(2)z^(3)).Apply Power Property: Apply the power property to the terms with exponents. Using the power property, we can write log(x^(5)) as 5 * log(x) and log(y^(2)z^(3)) as log(y^(2)) + log(z^(3)) because of the product inside the logarithm.Expand Product: Expand the logarithm of the product y^(2)z^(3). Using the product property of logarithms, which states that log_b(PQ) = log_b(P) + log_b(Q), we can write log(y^(2)z^(3)) as log(y^(2)) + log(z^(3)).Apply Power Property: Apply the power property to the remaining terms with exponents. Using the power property again, we can write log(y^(2)) as 2 * log(y) and log(z^(3)) as 3 * log(z).Combine Terms: Combine all the terms to get the final expanded form. Now we combine all the terms from the previous steps to get the final expanded form: 5 * log(x) - (2 * log(y) + 3 * log(z)).Distribute Negative Sign: Distribute the negative sign to the terms in the parentheses. Distributing the negative sign gives us 5 * log(x) - 2 * log(y) - 3 * log(z).

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{x^{5}}{y^{2} z^{3}}$ $\newline$ Answer:

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