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Which of the following statements is NOT correct? (A) log₁₀(1.4² - 1) is positive (B) The equation logₐ(a + 2) = 2 is satisfied by exactly two integral values of a (C) log_(0.1)(cot(3*pi/8)) is negative (D) If m = 4^(log₄(7)) and n = (1/9)^(-2*log₃(7)), then n = m⁴

1. log₁₀(1.4² - 1) is positive
2. the equation logₐ(a + 2) = 2 is satisfied by two integral values of a
3. log₀.1 cot(3*pi/8) is negative
4. if m = 4^(log₄ 7) and n = (1/9)^(-2*log₃ 7) then n = m⁴

Correct answer: log₁₀(1.4² - 1) is positive

Solution

1.4² - 1 = 0.96 < 1, so log₁₀(0.96) < 0, not positive — statement (A) is false. Statements (C): cot(67.5 deg) = tan(22.5 deg) ~ 0.414, log_(0.1)(0.414) = log(0.414)/log(0.1) = positive (both logs negative, ratio positive) — (C) is also incorrect, but (A) is the primary intended false statement. (D) is verifiably true: m=7, n=7⁴=m⁴.

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