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Chapter 13: Problem 42

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{9} \frac{g f^{2}}{h^{3}}$$

Short Answer

Expert verified
\(\log_{9}{g} + 2 \cdot \log_{9}{f} - 3 \cdot \log_{9}{h}\)

Step by step solution

01

Write the logarithm as a difference of logarithms using the quotient rule

The quotient rule states that \(\log_b{\frac{A}{B}} = \log_b{A} - \log_b{B}\). Applying this rule to our logarithmic expression, we get: \[\log _{9} \frac{g f^{2}}{h^{3}} = \log_{9}{(g f^2)} - \log_{9}{(h^3)}\]
02

Use the product rule to rearrange the terms

The product rule states that \(\log_b{(A \cdot B)} = \log_b{A} + \log_b{B}\). We will use this rule to separate the terms g and f^2 in the first logarithm: \[\log_{9}{(g f^2)} - \log_{9}{(h^3)} = \log_{9}{g} + \log_{9}{f^2} - \log_{9}{h^3}\]
03

Apply the power rule to simplify the logarithms

The power rule states that \(\log_b{(A^n)} = n \cdot \log_b{A}\). Applying this rule to the logarithms with f^2 and h^3, we get: \[\log_{9}{g} + \log_{9}{f^2} - \log_{9}{h^3} = \log_{9}{g} + 2 \cdot \log_{9}{f} - 3 \cdot \log_{9}{h}\] The given logarithmic expression has now been rewritten as a sum or difference of logarithms and simplified. The final answer is: \[\log_{9}{g} + 2 \cdot \log_{9}{f} - 3 \cdot \log_{9}{h}\]

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