# how to find the value of $\log_3 7$

Can I ask how to compute $\log_3 7$, using the changing the base of logarithm.

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$\log_3(7)$ is only an expression. One does not solve expressions. One solves equations like $\log_3(7) = x - 2$ (for example). Is something missing? – JavaMan May 2 '11 at 15:10
If you mean evaluate in terms of logarithms that appear on your calculator, you want to use the change of base formula to write it in terms of $\ln$ or $\log_{10}$. Do you know what the change of base formula is? – Jonas Meyer May 2 '11 at 15:14
@DJC: My guess is that the question is meant to be "how can I compute $\log_3(7)$" using the change of base of logarithm (e.g., if you only know how to compute natural or common logarithms...) – Arturo Magidin May 2 '11 at 15:14
sorry for my bad english, i should use find the values of instead – dramasea May 2 '11 at 15:15
This is a duplicate of at least a few other old questions. – Bill Dubuque May 2 '11 at 16:20

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 $$\log_3 7 = \frac{\log 7}{\log 3}.$$
@Hans: Heh; I honestly don't consider the change-of-base formula as a "basic property". To me, the "basic properties" are $\log_a(b) = r \Leftrightarrow a^r=b$ (definition), and the corresponding equalities that are derived from the basic properties of the exponential: $a^ra^s = a^{r+s}$ and $(a^r)^s = a^{rs}$ – Arturo Magidin May 2 '11 at 16:07