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Chapter 13: Problem 43

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{4} \frac{x^{3}}{y z^{2}}$$

Short Answer

Expert verified
The simplified expression as the sum or difference of logarithms is: \(3\log_4(x) - \log_4(y) - 2\log_4(z)\).

Step by step solution

01

Applying the Quotient Rule

The Quotient Rule states that \(\log_b(\frac{p}{q}) = \log_b(p) - \log_b(q)\). We will apply this rule to our given expression: \(\log_4(\frac{x^3}{yz^2}) = \log_4(x^3) - \log_4(yz^2)\)
02

Applying the Product Rule

The Product Rule states that \(\log_b(pq) = \log_b(p) + \log_b(q)\). We will apply this rule to the second part of our expression: \(\log_4(yz^2) = \log_4(y) +\log_4(z^2)\) So our expression now becomes: \(\log_4(\frac{x^3}{yz^2}) = \log_4(x^3) - (\log_4(y) + \log_4(z^2))\)
03

Distributing the Negative Sign

Now, distribute the negative sign to both terms inside the parentheses: \(\log_4(\frac{x^3}{yz^2}) = \log_4(x^3) - \log_4(y) - \log_4(z^2)\)
04

Applying the Power Rule

The Power Rule states that \(\log_b(p^q) = q\log_b(p)\). We'll apply this rule to both the \(\log_4(x^3)\) and \(\log_4(z^2)\) terms: \(\log_4(x^3) = 3\log_4(x)\) and \(\log_4(z^2) = 2\log_4(z)\) Now, we can substitute these back into our expression: \(\log_4(\frac{x^3}{yz^2}) = 3\log_4(x) - \log_4(y) - 2\log_4(z)\) The given expression has been simplified as a sum or difference of logarithms. The final answer is: \(3\log_4(x) - \log_4(y) - 2\log_4(z)\)

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