<s>[INST] Sort those 3 logarithmic values without using calculator I found this problem interesting, namely we are given three values: $$\log_{1/3}{27}, \log_{1/5}{4}, \log_{1/2}{5}$$
We want to sort those values without using calculator. So I decided to work on them a little bit before comparing anything. Here is what I got:
First of all: $\log_{1/3}{27} = \log_{1/3}{3^3} = \log_{1/3}{(\frac{1}{3})^{-3}} = -3\\ \log_{1/5}{4} = \log_{1/5}{2^2} = 2 \cdot \log_{1/5}{2} = 2 \cdot \log_{5^{-1}}{2} = -2 \cdot \log_{5}{2}$
Since $\log_{5}{2} < 1, \text{then: } -2\cdot \log_{5}{2} > -2 $
And the third value: $\log_{1/2}{5} = \log_{2^{-1}}{5} = -\log_{2}{5}$
Since $2 < \log_{2}{5} < 3, \text{then: } -2 > -\log_{2}{5} > -3 $
This shows us that the first values is the smallest, then the third one and finally the second one: $$\log_{1/3}{27}, \log_{1/2}{5}, \log_{1/5}{4}$$
Is my approach correct or have I made some mistakes, can we get more accurate values for those logarithms without using calculator.
 [/INST] As an alternative and to check recall that
$$\log_b a=\frac{\log_c a}{\log_c b}$$
that is
$$\log_{\frac13}27=\frac{\log_2 27}{\log_2\frac13}=-\frac{\log_2 3^3}{\log_2 3}=-3$$
$$\log_{\frac15}4=\frac{\log_2 4}{\log_2 \frac15}=-\frac{\log_2 2^2}{\log_2 5}=-\frac{2}{\log_2 5}$$
$$\log_{\frac12}5=\frac{\log_2 5}{\log_2 \frac12}=-\frac{\log 5}{\log_2 2}=-\log_2 5$$
therefore as you stated
$$\log_{\frac13}27<\log_{\frac12}5<\log_{\frac15}4$$
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