Establish trichotomy in each of this following pairs of numbers from the attachment given below. Explain.
Answers
Given : 3^(log₂₇3) and 2^(log₄2 )
To find : Establish trichotomy ( relation of smaller , larger , equal)
Solution:
log₂₇3 = log 3/log 27 = log 3/ log 3³ = log 3/ 3log 3 = 1/3
log₄2 = log 2/log 4 = log 2/ log 2² = log 2/ 2log 2 = 1/2
∛3 = 1.44
√2 = 1.41
3^(log₂₇3) > 2^(log₄2 )
log₄5 = log 5/log 4
log₁/₁₆ (1/25) = log (1/25)/log (1/16) = log (5)⁻²/log (4)⁻² = -2log5/-2log4
= log 5 /log 4
Hence log₄5 = log₁/₁₆ (1/25)
log₈₁10 + log₁₀81
= log10/log81 + log81/log10
= 1 /log 3⁴ + log 3⁴/log 10
= 1/4log3 + 4 log3
log3 = 0.477
= 0.524 + 1.91
= 2.434 < 4
4 > log₈₁10 + log₁₀81
log (1/7) /log (1/5) = log 7⁻¹ / log 5⁻¹ = log 7 / log 5
log (1/5) /log (1/7) = log 5⁻¹ / log 7⁻¹ = log 5 / log 7
log 7 / log 5 > 1
log 5 / log 7 < 1
log 7 / log 5 > log 5 / log 7
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