$\log_{10} \log_{10} \log_{10} (10^{10^{10}})$ is equal to

Concept & Formula This question deals with nested logarithms operating on a power tower (an iterated exponentiation). The key property to apply repeatedly is: $$\log_a (a^x) = x$$ This rule allows the outermost exponent to "drop down" when the base of the logarithm matches the base of the exponential term. Step-by-Step Solution * **Given:** $\log_{10} [ \log_{10} [ \log_{10} (10^{10^{10}}) ] ]$ * **Evaluate Innermost Logarithm:** Apply the rule $\log_{10}(10^x) = x$ to the innermost logarithm. Here, the exponent $x$ is the upper part of the tower, $10^{10}$. $$\log_{10} (10^{10^{10}}) = 10^{10}$$ * **Substitute and Evaluate Second Logarithm:** Now, the expression becomes: $$\log_{10} [ \log_{10} (10^{10}) ]$$ * Apply the same rule again. Ask: "What is the exponent on the base 10?" The exponent is 10. $$\log_{10} (10^{10}) = 10$$ * **Substitute and Evaluate Outermost Logarithm:** The expression is now reduced to a single logarithm: $$\log_{10} (10)$$ * **Final Calculation:** Since $10^1 = 10$, this evaluates to 1. $$1$$ Exam Strategy & Shortcut Each time you take $\log_{10}$ of a power of 10, you simply chop off the bottom "10" of the exponent tower. There are three $\log_{10}$ operations and three "10"s in the exponent tower $10^{10^{10}}$. Operation 1 knocks out the base 10 leaving $10^{10}$. Operation 2 knocks out the next 10 leaving 10. Operation 3 knocks out the final 10 leaving an implied exponent of 1. Therefore, they cancel out perfectly down to 1. Common Pitfall Students frequently misinterpret $10^{10^{10}}$ as $(10^{10})^{10}$, which would be $10^{100}$. A power tower is always evaluated from the top down, so $10^{10^{10}}$ means $10^{(10^{10})}$. Misinterpreting this can cause you to apply the log rules incorrectly and get stuck with an extra factor. Final Answer **Therefore, the correct answer is 1.**