Given log10 57 ≈ 1.756 and log10 0.57 = log10(57) − 2 ≈ −0.244, evaluate: log 57 + log(0.57)^3 + log √(0.57).

Introduction / Context:We combine logs using properties: log(a^k) = k log a and log a + log b = log(ab). The expression contains powers of the same positive number (0.57), which we merge systematically.

Given Data / Assumptions:

• log10 57 ≈ 1.756.
• log10 0.57 = log10 57 − 2 ≈ −0.244.
• Expression: log 57 + log(0.57)^3 + log √(0.57).

Concept / Approach:

• Interpret log(0.57)^3 as log((0.57)^3) = 3 log(0.57).
• Similarly, log √(0.57) = (1/2) log(0.57).

Step-by-Step Solution:

Verification / Alternative check:Combine inside a single log: S = log[57 * (0.57)^{3.5}] and evaluate numerically; the same value results to three decimals.

Why Other Options Are Wrong:

• 1.902 and 2.146 add instead of subtracting the negative contributions.
• 1.146 is from rounding incorrectly.

Common Pitfalls:

• Misreading log(0.57)^3 as (log 0.57)^3; here the intended property is log(a^k).

Final Answer:0.902