\( om!\log _{10}({}_{\log _{10}(\log _{10}(\log _{10}({}_{\log _{10}(e_{})}P_{})))}P_{}) \)

It seems like there is a bit of confusion in the notation of your problem. The expression you've provided is not well-formed, and it's difficult to interpret what you're asking. However, I can try to help you with a few different interpretations. If you're looking for the logarithm of a logarithm, the general form would be: \[ \log_{10}(\log_{10}(x)) \] where \( x \) is the argument inside the inner logarithm. If you're trying to apply this to a specific value, such as \( e \), the expression would be: \[ \log_{10}(\log_{10}(e)) \] This expression can be simplified as follows: 1. The logarithm of \( e \) to the base \( 10 \) is \( \log_{10}(e) \). Since \( e \) is the base of the natural logarithm, \( \log_{10}(e) \) is approximately \( 1 \) because \( 10^1 = e \). 2. The logarithm of \( 1 \) to the base \( 10 \) is \( 0 \) because \( 10^0 = 1 \). So, the simplified form of the expression is: \[ \log_{10}(\log_{10}(e)) = \log_{10}(1) = 0 \] If you meant something different or if there are additional details or corrections to the expression, please provide them so I can assist you further.