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Hello everyone. I'm stuck on this problem and not sure how to apply the Unique factorization therem also called (Fundamental Theorem of Arithmetic). Heres the problem:

http://suprfile.com/src/1/45ym0fx/lastscan.jpg [Broken]

The back of the book gives a pretty big hint which is the following:

#20. HINT: Use a prpoof by contraction. Suppose log_3 (7) is rational. Then log_3 (7) = a/b for some integers a and b with b != 0. Apply the definition of logarithm to rewrite log_3 (7) = a/b in exponential form.

I know 3 and 7 are prime numbers, but I'm confused on what I'm suppose to do with the (a/b), usually when I prove by contradiction with rational, i solve for the part that they claimed was irrational, and showing a and b are intgers proves its rational which contradicts but in this case I'm lost.

Could I say, Since (a/b) is an integer, which satisfies e_1, and 7 is also an integer, so that would be n, and 3 is a prime number, so that would be p_1...but i don't see how this is proving its rational but it does fit the Unique factorization theorem.

And the problem is saying use logarithm and also the Unique factorization theroem to prove this. I'm stuck on how to apply the unique factorization part. Any help would be great!

http://suprfile.com/src/1/45ym0fx/lastscan.jpg [Broken]

The back of the book gives a pretty big hint which is the following:

#20. HINT: Use a prpoof by contraction. Suppose log_3 (7) is rational. Then log_3 (7) = a/b for some integers a and b with b != 0. Apply the definition of logarithm to rewrite log_3 (7) = a/b in exponential form.

**Note: the unique factorization theorem states:**

GIven any integer n > 1, there exists a postive integer k, disticnt prime numbers p1, p2, ... pk and postive intgers e_1,e_2...,e_k such that

n = p_1^(e_1)*p_2^(e_2)*p_3^(e_3)...p_k^(e_k)

and any other expression of n as a product of prime numbers id idneitical to this except, perhaps, for the ordder in which the factors are writtenGIven any integer n > 1, there exists a postive integer k, disticnt prime numbers p1, p2, ... pk and postive intgers e_1,e_2...,e_k such that

n = p_1^(e_1)*p_2^(e_2)*p_3^(e_3)...p_k^(e_k)

and any other expression of n as a product of prime numbers id idneitical to this except, perhaps, for the ordder in which the factors are written

I know 3 and 7 are prime numbers, but I'm confused on what I'm suppose to do with the (a/b), usually when I prove by contradiction with rational, i solve for the part that they claimed was irrational, and showing a and b are intgers proves its rational which contradicts but in this case I'm lost.

Could I say, Since (a/b) is an integer, which satisfies e_1, and 7 is also an integer, so that would be n, and 3 is a prime number, so that would be p_1...but i don't see how this is proving its rational but it does fit the Unique factorization theorem.

And the problem is saying use logarithm and also the Unique factorization theroem to prove this. I'm stuck on how to apply the unique factorization part. Any help would be great!

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