Compare log_3 4 and 2^(1/4)

Jack Ingram

Jack Ingram

Answered question

2022-10-31

Compare log 3 4 and 2 1 4
Both log 3 4 and 2 1 4 are somewhere in between 1 and 2. I know I can get approximate values but they are difficult to calculate by hand so I conclude it's not the way I'm supposed to do the exercise.
So how can I compare the two numbers without using a calculator?

Answer & Explanation

Sauppypefpg

Sauppypefpg

Beginner2022-11-01Added 23 answers

Since ( 5 4 ) 4 = 625 256 > 512 256 = 2, we have 2 4 < 5 4 . Hence its enough to show 5 4 < log 3 4. This is equivalent to 4 > 3 5 4 , 4 4 > 3 5 , 256 > 243 which is true. So 2 4 < 5 4 < log 3 4.

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