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Chapter 13: Problem 13

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{9} \frac{4}{7}$$

Short Answer

Expert verified
Using the quotient rule for logarithms, the given expression can be simplified as follows: \(\log_9\frac{4}{7} = \log_9{4} - \log_9{7}\).

Step by step solution

01

Identify the quotient rule for logarithms

The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator: \(\log_b{\frac{M}{N}} = \log_b{M} - \log_b{N}\) In our case, the base is 9, the numerator is 4, and the denominator is 7.
02

Apply the quotient rule to the given logarithm

Applying the quotient rule to the given logarithm, we obtain: \(\log_9\frac{4}{7} = \log_9{4} - \log_9{7}\) Now, we've expressed the logarithm as the difference of two simpler logarithms. We don't have any further simplification, since there's no property that simplifies the expressions of \(\log_9{4}\) and \(\log_9{7}\) in base 9. So, the final answer is: \(\log_9\frac{4}{7} = \log_9{4} - \log_9{7}\)

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