Evaluate log x + log(1/x) by combining into a single logarithm. State the exact value.

Introduction / Context:We evaluate a basic identity involving a variable and its reciprocal inside logarithms of the same base.

Given Data / Assumptions:

• x > 0
• Expression: log x + log(1/x)

Concept / Approach:Use log A + log B = log(AB). Since x · (1/x) = 1, the expression reduces to log(1).

Step-by-Step Solution:

Verification / Alternative check:Regardless of base (consistent across both logs), log(1) equals 0, so the value is unambiguously 0.

Why Other Options Are Wrong:1 or −1 would correspond to log(10) or log(0.1) in base 10, but the argument here equals 1; 1/2 is likewise irrelevant.

Common Pitfalls:Attempting to add arguments rather than multiply (x + 1/x) is a different expression and not applicable here.

Final Answer:0