it is my exercise to compute the numerical value of $x$ and $y$, where $$x:=\log_{20}(100)+\log_{100}(20),\\
y:=\log_{\frac{1}{2}}(70)+\log_{\frac{1}{10}}(200)+\log_{\frac{3}{2}}\frac{1}{100}.$$
In order to do so, I tried to simplify the expressions in various ways (by using the usual rules of logarithm), but I can not do it in a satisfactory way, such that the values are obvious. Is there any trick you can tell me which might work?
Let my show you, for example, what I tried to find $x$ explicitly. First I observed that the terms $\log_{20}(100)$ and $\log_{100}(20)$ do not seem to be computable individually. As I said before, I tried then to manipulate it in various ways, but I always get stuck, e.g.:
$$x= \log_{20}(100) + \frac{1}{\log_{20}(100)}=?\\
x= 2 \log_{20}(5)+ 2\log_{20}(2)+ 2\log_{100}(2)+ \log_{100}(5) = ?$$
etc.
I would appreciate your help very much!
Best
Best Answer
There is a formula to change the base of logarithms $$\log_a b=\frac{\log_c b}{\log_c a}$$ for instance $$\log_{20} 100+\log_{100} 20=\frac{\log_{100} 100}{\log_{100} 20}+\log_{100} 20=\frac{1}{\log_{100} 20}+\log_{100} 20$$ Note that $$\log_{100}x=\frac{\log_{10}x}{\log_{10}100}=\frac{1}{2}\log_{10}x$$