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.

Question

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.

diamondogsaz · Accepted Answer

You can&#039;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  ⁡  3that becomes  (  2  b  −  a  )  ln  ⁡  2  =  (  2  a  −  b  )  ln  ⁡  3which 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&#039;s impossible because of unique factorization of integers.

Jonathan Miles · Accepted Answer

a      b        =                    ln        ⁡                  12                            ln        ⁡                  18                      =      log          18        ⁡          12      ⟺                                    18                      a                          /                        b                                              =        12                                      18          a                                    =                  12          b

How to prove that ln ⁡ 12 </mrow>

Answered question

Answer & Explanation

New Questions in Pre-Algebra