高一對數問題 求計算過程?

13. 化簡 log₈ (√(53 + 10√6) + √(21 - 6√6)) = ____
解:
log₈ (√(53 + 10√6) + √(21 - 6√6))
= log₈ (√(50 + 2(5√2)(√3) + 3) + √(18 - 2(3√2)(√3) + 3))
= log₈ (√(5√2 + √3)² + √(3√2 - √3)²)
= log₈ ( 5√2 + √3 + 3√2 - √3)
= log₈ 8√2 = log₈ 8(8^(1/6)) = log₈ 8^(7/6) = 7/6.


14. 設 x = 1/√(3+√8) , 則 log₈(x⁴+ 4x³ + 4x² + 4x + 3) = ____
解:
x² = 1/(3+√8) = 3 - √8 = 2 - 2√2 + 1 = (√2 - 1)²
x = √2 - 1
x² + 2x - 1 = 0
則 x⁴+ 4x³ + 4x² + 4x + 3 = x²(x² + 4x + 4) + 4x + 3 = x²(2x + 5) + 4x + 3
= x(2x² + 5x + 4) + 3 = x(x + 6) + 3 = x² + 6x + 3 = 4x + 4 = 4√2 ,
log₈ 4√2 = log₈ √2^5 = log₈ (√2^6)^(5/6) = log₈ 8^(5/6) = 5/6.