Questions: Rewrite the logarithm as a ratio of common logarithms and natural logarithms. log9(35) (a) common logarithms (b) natural logarithms

Solution Steps

To rewrite the logarithm \(\log_{9}(35)\) using common logarithms (base 10) and natural logarithms (base \(e\)), we can use the change of base formula. The change of base formula states that \(\log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)}\), where \(k\) is the new base. For common logarithms, \(k = 10\), and for natural logarithms, \(k = e\).

Using the change of base formula, we can express \(\log_{9}(35)\) in terms of common logarithms (base 10) as follows: \[ \log_{9}(35) = \frac{\log_{10}(35)}{\log_{10}(9)} \] Calculating this gives: \[ \log_{9}(35) \approx 1.6181 \]

Similarly, we can express \(\log_{9}(35)\) in terms of natural logarithms (base \(e\)): \[ \log_{9}(35) = \frac{\ln(35)}{\ln(9)} \] Calculating this also yields: \[ \log_{9}(35) \approx 1.6181 \]

Final Answer