Given log 10(3) = 0.477 and (1000)^x = 3, find x by applying change-of-base concepts to exponential–log relations.

Introduction / Context:We relate an exponential equation to logarithms by taking the common logarithm of both sides or by comparing powers of 10 directly.

Given Data / Assumptions:

• (1000)^x = 3
• log 10(3) = 0.477 (approximate)

Concept / Approach:Since 1000 = 10^3, we have (10^3)^x = 10^{3x}. Equate exponents via logs: 10^{3x} = 3 ⇒ 3x = log 10(3) ⇒ x = log 10(3)/3.

Step-by-Step Solution:

Verification / Alternative check:10^{0.477} ≈ 3, so 10^{3·0.159} ≈ 10^{0.477} ≈ 3, confirming the value of x.

Why Other Options Are Wrong:Other decimals correspond to dividing by 10 or 100 instead of 3, or to unrelated operations; 10 is clearly far outside the correct scale.

Common Pitfalls:Forgetting that 1000 = 10^3 and using base e without adjusting leads to numerical mistakes; the ratio approach avoids these slips.

Final Answer:0.159