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logiski9s 2022-07-03 Answered
How to prove that ln 12 ln 18 is irrational witout using the change of base rule?
I have to show that ln 12 ln 18 is irrational by using change of base rule.
At the beginning I have proved that ln r is irrational for any rational r, r 1. Then using this we can say that ln 12 and ln 18 are irrational.
But from here it is difficult for me to show that the fraction is irrational knowing that both the numerator and the denominator are irrational.
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Answers (2)

diamondogsaz
Answered 2022-07-04 Author has 12 answers
You can't prove it using that the quotient of irrationals is irrational for the simple reason that the statement is false.
You may instead use a different strategy: suppose
ln 12 ln 18 = 2 ln 2 + ln 3 ln 2 + 2 ln 3 = a b
for positive and coprime integers a and b. Then
2 b ln 2 + b ln 3 = a ln 2 + 2 a ln 3
that becomes
( 2 b a ) ln 2 = ( 2 a b ) ln 3
which tells you that ln 3 / ln 2 is rational as well. By the change of base rule, this is the same as saying that log 2 3 is rational, so
log 2 3 = h k
for positive integers h and k. Therefore
3 = 2 h / k
or
3 k = 2 h
that's impossible because of unique factorization of integers.

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Jonathan Miles
Answered 2022-07-05 Author has 3 answers
a b = ln 12 ln 18 = log 18 12 18 a / b = 12 18 a = 12 b

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