logiski9s

2022-07-03

How to prove that $\frac{\mathrm{ln}12}{\mathrm{ln}18}$ is irrational witout using the change of base rule?

I have to show that $\frac{\mathrm{ln}12}{\mathrm{ln}18}$ is irrational by using change of base rule.

At the beginning I have proved that $\mathrm{ln}r$ is irrational for any rational $r$, $r\ne 1$. Then using this we can say that $\mathrm{ln}12$ and $\mathrm{ln}18$ 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.

I have to show that $\frac{\mathrm{ln}12}{\mathrm{ln}18}$ is irrational by using change of base rule.

At the beginning I have proved that $\mathrm{ln}r$ is irrational for any rational $r$, $r\ne 1$. Then using this we can say that $\mathrm{ln}12$ and $\mathrm{ln}18$ 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

Beginner2022-07-04Added 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

$\frac{\mathrm{ln}12}{\mathrm{ln}18}=\frac{2\mathrm{ln}2+\mathrm{ln}3}{\mathrm{ln}2+2\mathrm{ln}3}=\frac{a}{b}$

for positive and coprime integers a and b. Then

$2b\mathrm{ln}2+b\mathrm{ln}3=a\mathrm{ln}2+2a\mathrm{ln}3$

that becomes

$(2b-a)\mathrm{ln}2=(2a-b)\mathrm{ln}3$

which tells you that $\mathrm{ln}3/\mathrm{ln}2$ is rational as well. By the change of base rule, this is the same as saying that ${\mathrm{log}}_{2}3$ is rational, so

${\mathrm{log}}_{2}3=\frac{h}{k}$

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.

You may instead use a different strategy: suppose

$\frac{\mathrm{ln}12}{\mathrm{ln}18}=\frac{2\mathrm{ln}2+\mathrm{ln}3}{\mathrm{ln}2+2\mathrm{ln}3}=\frac{a}{b}$

for positive and coprime integers a and b. Then

$2b\mathrm{ln}2+b\mathrm{ln}3=a\mathrm{ln}2+2a\mathrm{ln}3$

that becomes

$(2b-a)\mathrm{ln}2=(2a-b)\mathrm{ln}3$

which tells you that $\mathrm{ln}3/\mathrm{ln}2$ is rational as well. By the change of base rule, this is the same as saying that ${\mathrm{log}}_{2}3$ is rational, so

${\mathrm{log}}_{2}3=\frac{h}{k}$

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.

Jonathan Miles

Beginner2022-07-05Added 3 answers

$\frac{a}{b}}={\displaystyle \frac{\mathrm{ln}12}{\mathrm{ln}18}}={\mathrm{log}}_{18}\phantom{\rule{negativethinmathspace}{0ex}}12\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\begin{array}{rl}{18}^{a/b}& =12\\ {18}^{a}& ={12}^{b}\end{array$

Which expression has both 8 and n as factors???

One number is 2 more than 3 times another. Their sum is 22. Find the numbers

8, 14

5, 17

2, 20

4, 18

10, 12Perform the indicated operation and simplify the result. Leave your answer in factored form

$\left[\frac{(4x-8)}{(-3x)}\right].\left[\frac{12}{(12-6x)}\right]$ An ordered pair set is referred to as a ___?

Please, can u convert 3.16 (6 repeating) to fraction.

Write an algebraic expression for the statement '6 less than the quotient of x divided by 3 equals 2'.

A) $6-\frac{x}{3}=2$

B) $\frac{x}{3}-6=2$

C) 3x−6=2

D) $\frac{3}{x}-6=2$Find: $2.48\xf74$.

Multiplication $999\times 999$ equals.

Solve: (128÷32)÷(−4)=

A) -1

B) 2

C) -4

D) -3What is $0.78888.....$ converted into a fraction? $\left(0.7\overline{8}\right)$

The mixed fraction representation of 7/3 is...

How to write the algebraic expression given: the quotient of 5 plus d and 12 minus w?

Express 200+30+5+4100+71000 as a decimal number and find its hundredths digit.

A)235.47,7

B)235.047,4

C)235.47,4

D)234.057,7Find four equivalent fractions of the given fraction:$\frac{6}{12}$

How to find the greatest common factor of $80{x}^{3},30y{x}^{2}$?