Infinite logarithm

$\large\log_{10\color{#3D99F6}{\log_{10} 10}} 10\\ \large=\log_{10 \color{#3D99F6}{(1)}} 10\\ \large=\log_{10} 10\\ \large=1$

$\Large \log_{10\log_{10\color{#3D99F6}{\log_{10} 10}} 10} 10\\ \Large =\log_{10 \log_{10\color{#3D99F6}{(1)}}10}10\\ \Large= \log_{10 \color{#3D99F6}{\log_{10} 10}} 10\\ \Large=\log_{10 \color{#3D99F6}{(1)}} 10\\ \Large=\log_{10} 10\\ \Large=1$

$\Large \log_{10 \log_{10 \log_{10\color{#3D99F6}{\log_{10} 10}} 10} 10} 10\\ \Large = \log_{10 \log_{10 \log_{10\color{#3D99F6}{(1)}} 10} 10} 10\\ \Large =\log_{10\log_{10\color{#3D99F6}{\log_{10} 10}} 10} 10\\ \Large =\log_{10 \log_{10\color{#3D99F6}{(1)}}10}10\\ \Large= \log_{10 \color{#3D99F6}{\log_{10} 10}} 10\\ \Large=\log_{10 \color{#3D99F6}{(1)}} 10\\ \Large=\log_{10} 10\\ \Large=1$

From here, you can see a pattern. No matter how many times this logarithm is repeated, you will get the same answer. You can continue testing out for 5 layers and above, and the pattern will still hold true.

Therefore, we can say that

$\huge \log_{10 \log_{10 \log_{10 \log_{\ddots} 10 }10}10}10 = \boxed{1}$

Suppose the equation is equal to x. So, change in exponent form which gives $(10x)^x = 10$ , which holds only for $x = 1$ .