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Chapter 13: Problem 41

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

Short Answer

Expert verified
The short answer for the given problem is: \(\log _{2} \frac{4 \sqrt{n}}{m^{3}} = 2 + \frac{1}{2}\log_{2}(n) - 3\log_{2}(m)\)

Step by step solution

01

Apply the Quotient Rule

Recall the Quotient Rule of logarithms: \(\log_{a}(\frac{b}{c}) = \log_{a}(b) - \log_{a}(c)\). Using this rule, we can rewrite the given expression as follows: \(\log _{2} \frac{4 \sqrt{n}}{m^{3}} = \log_{2}(4 \sqrt{n}) - \log_{2}(m^{3})\)
02

Apply the Product Rule

Now, we see that a product is the argument of the first logarithm. Recall the Product Rule of logarithms: \(\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)\). Using this rule, we can rewrite the first logarithm as follows: \(\log_{2}(4 \sqrt{n}) = \log_{2}(4) + \log_{2}(\sqrt{n})\)
03

Apply the Power Rule

Both remaining logarithms have exponents in their arguments. Recall the Power Rule of logarithms: \(\log_{a}(b^c) = c\log_{a}(b)\). Using this rule, we can rewrite the logarithms as follows: \(\log_{2}(\sqrt{n}) = \frac{1}{2}\log_{2}(n)\) and \(\log_{2}(m^{3}) = 3\log_{2}(m)\)
04

Combine and Simplify

Now, we can substitute the expressions from Steps 2 and 3 back into the expression from Step 1, then simplify: \(\log _{2} \frac{4 \sqrt{n}}{m^{3}} = (\log_{2}(4) + \frac{1}{2}\log_{2}(n)) - 3\log_{2}(m)\) Since \(\log_{2}(4) = 2\), the simplified expression is: \(2 + \frac{1}{2}\log_{2}(n) - 3\log_{2}(m)\)

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