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Chapter 3: Problem 63

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{5} \frac{x^{2}}{y^{2} z^{3}}$$

Short Answer

Expert verified
The expanded expression of \(\log_5 \frac{x^2}{y^2 z^3}\) is \(2 \log_5 x - 2 \log_5 y - 3 \log_5 z\).

Step by step solution

01

Apply the Quotient Rule of Logarithms

We can use the quotient rule which states that \(\log_b \frac{M}{N} = \log_b M - \log_b N\). Here, M = \(x^2\) and N = \(y^2 z^3\). When we apply the quotient rule to \(\log_5 \frac{x^2}{y^2 z^3}\), we get \(\log_5 x^2 - \log_5 y^2 z^3\).
02

Separate the Subtracting Terms

Next, we need to separate the subtracting terms. We use the logarithm rule \(\log_b MN = \log_b M + \log_b N\), and apply it to the second term \(\log_5 y^2 z^3\), we get \(\log_5 x^2 - (\log_5 y^2 + \log_5 z^3)\).
03

Apply the Power Rule of Logarithms

Finally, we take the power rule into account, which states that \(\log_b M^n = n \log_b M\). Applying this rule to each term in the expression we got in the previous step, we end up with \(2 \log_5 x - 2 \log_5 y - 3 \log_5 z\).

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Most popular questions from this chapter

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