Can I ask how to compute $\log_3 7$, using the changing the base of logarithm.
how to find the value of $\log_3 7$
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logarithms
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2This is a duplicate of at least a few other old questions. – 2011-05-02
1 Answers
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If you mean, "How can I calculate $\log_3 7$ using the change of base formula?":
I've never memorized the change of base formula, I always re-derive it as needed. The key is to remember what the expression means: $\log_3 7 = r$ means that $3^r = 7$. Taking logarithms base $b$ on both sides, we have $\begin{align*} 3^r &= 7\\ \log_b(3^r) &= \log_b(7)\\ r\log_b 3&= \log_b 7\\ r &= \frac{\log_b 7}{\log_b 3}\\ \log_3 7 &= \frac{\log_b 7}{\log_b 3}. \end{align*}$ So if you want to compute $\log_3 7$ using the natural log, you would have $\log_3 7 = \frac{\ln 7}{\ln 3}.$ If you want to compute them using the common logarithm (base 10), you would compute $\log_3 7 = \frac{\log 7}{\log 3}.$
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0All the same, from experience, I've found that each of these things are worth deriving in front of students needing help. Logs seem to be especially mysterious to even competent calculus students. The derivations can help demystify. – 2011-05-02