What is the value of [log10 (5log10 100)]2 = ?

Concept: If bn = x, then logb(x) = n; equivalently, k times logb(x) = logb(xk) - the power rule of logarithms. These two identities let a compound logarithmic expression be reduced step by step from the innermost bracket outward.

1. Evaluate the innermost logarithm: log10(100). Since 102 = 100, log10(100) = 2.

2. Substitute into the multiplier: 5 times log10(100) = 5 times 2 = 10.

3. The expression now reads [log10(10)]2.

4. Evaluate log10(10): since 101 = 10, log10(10) = 1.

5. Square this result: 12 = 1.

Cross-check: Using the power rule instead: 5 log10(100) = log10(1005) = log10(1010) = 10, the same intermediate value as before; squaring log10(10) = 1 again gives 12 = 1, confirming the result independently.

Result: The value of the expression is 1.