# Solving equation $\log_y(\log_y(x))= \log_n(x)$ for $n$

I'm just wondering, if I log a constant twice with the same base $y$,

$$\log_y(\log_y(x))= \log_n(x)$$

Can it be equivalent to logging the same constant with base $n$? If yes, what is variable $n$ equivalent to?

• In light of your comment to Avatar, you might be interested in my post Exponential and Logarithmic Commutativity. Mar 6, 2013 at 17:22
• @DaveL.Renfro Your post is pretty inspirational. Thanks for sharing! I've learnt a new approach to solving log problems from it. When I have time I would definitely look into question 5 and 6 and try to find a solution for them. They look fun. Mar 10, 2013 at 13:53
• If you send me an email (I can't find your email address), I can send you a .pdf file of my original submitted "article". The .pdf version I have is a little more complete than the published version (because of space considerations), which in turn is longer than what I posted. Click on my name to find my email address. Mar 11, 2013 at 17:02

No - take as example, $x = 1$. Then $\log_y(x) = 0$ and $\log_y(\log_y(x))$ is undefined. However as $x = 1$, $\log_n(x)$ is always equal to $0$, which means that $\log_y(\log_y(x)) \neq \log_n(x)$

• $x=y$ is a similarly nice example. Mar 5, 2013 at 8:31

$$\log_y(\log_y(x))= \log_n(x)$$ $$\implies \log_y(\log_y(x))= \frac{\log_e(x)}{\log_e(n)}$$ $$\implies \log_e(n)=\frac{\log_e(x)}{\log_y(\log_y(x))}$$ $$\implies n=e^{\frac{\log_e(x)}{\log_y(\log_y(x))}}$$

Thus, for given $x,y$, if $e^{\frac{\log_e(x)}{\log_y(\log_y(x))}}$ is defined, then that is the value for $n$.

• Bravo! I have once again witnessed the beauty of mathematics. Mar 5, 2013 at 11:32

Yes you can do that with the initial conditions for a logarithm satisfying.

The conditions for log(x) to the base n are: x > 0, n > 0 and n != 1.

so you should be careful with the domain that you are choosing.

• i think my example is one reason of this Mar 5, 2013 at 8:24
• Yes, but you have to stick to the domain of the function as well.
– lsp
Mar 5, 2013 at 10:33

suppose $x=4$ and $y=2$ then $\log_y(x)=2$ and $\log_2(2)=1$

which means that $\log_n(4)=1$ it means that $n=4$. I don't know if it is helpful for you.