Evaluate the expression (1 / 3) log10 125 - 2 log10 4 + log10 32.

Introduction / Context:
This question focuses on evaluating an expression involving common logarithms by applying standard log properties such as the power rule and the product and quotient rules. It helps students practise simplifying log expressions without resorting to calculators. These skills are very important in aptitude tests and competitive exams that emphasise algebraic manipulation.

Given Data / Assumptions:
- The expression is E = (1 / 3) log10 125 - 2 log10 4 + log10 32.
- All logarithms are common logarithms (base 10).
- The numbers 125, 4 and 32 are positive, so the logs are defined.

Concept / Approach:
We use three basic properties of logs:
1. Power rule: k log10 A = log10 (A^k).
2. Sum rule: log10 A + log10 B = log10 (A * B).
3. Difference rule: log10 A - log10 B = log10 (A / B).
By rewriting each term using the power rule, we can combine all terms into a single log10 of a simple number. Then we evaluate that number directly to find the final value.

Step-by-Step Solution:
Start with E = (1 / 3) log10 125 - 2 log10 4 + log10 32. Apply the power rule to the first term: (1 / 3) log10 125 = log10 (125^(1 / 3)). Since 125 = 5^3, its cube root 125^(1 / 3) is 5, so (1 / 3) log10 125 = log10 5. Apply the power rule to the second term: -2 log10 4 = log10 (4^-2). Compute 4^-2 = 1 / 4^2 = 1 / 16. So the expression becomes E = log10 5 + log10 32 + log10 (4^-2). Use the sum rule to combine: E = log10 [5 * 32 * 4^-2]. Compute 4^2 = 16, so 4^-2 = 1 / 16. Thus E = log10 [5 * 32 / 16]. Calculate the product and quotient: 32 / 16 = 2, so 5 * 2 = 10. Hence E = log10 10. We know log10 10 = 1.

Verification / Alternative check:
We can also express everything in terms of powers of 2 and 5. Note that 125 = 5^3, 4 = 2^2, and 32 = 2^5. Rewriting E as (1 / 3) log10 (5^3) - 2 log10 (2^2) + log10 (2^5) and applying the power rule directly gives log10 5 - 4 log10 2 + 5 log10 2 = log10 5 + log10 2 = log10 10 = 1. This matches our previous result, confirming the answer.

Why Other Options Are Wrong:
0: This would require the combined product inside the final logarithm to be 1, which it is not.
2: This would correspond to log10 of 100, but we found that the effective argument is 10, not 100.
4/5: This is not consistent with any simple power of 10 and does not match the simplified log expression.

Common Pitfalls:
Students sometimes forget to apply the power rule correctly and treat (1 / 3) log10 125 as log10 (125 / 3), which is wrong. Another common error is mishandling negative exponents and incorrectly simplifying 4^-2. Misapplication of the sum and difference rules can also lead to combining terms in the wrong way. Keeping the three basic log rules clear and working step by step avoids these problems.

Final Answer:
The value of the expression is 1.