What is the meaning of this Wolfram Alpha result when calculating 3^p=4^q? I would like to know are the some p in bbbN and q in bbbN for 3^p=4^q except the trivial p=q=0. So, I entered the expression into Wolfram Alpha, which returned the result for q: q=(log(3^p))/(\log(4))+(2 pi i c_(1))/(\log(4)) Nothing is said about what is c_1. Arbitrary constant? If c_1!=0 the result could be complex-valued I guess. I tried some calculations using the formula but the results do not make sense.

fermetrakr

fermetrakr

Open question

2022-08-30

What is the meaning of this Wolfram Alpha result when calculating 3 p = 4 q ?
I would like to know are the some p N and q N for 3 p = 4 q except the trivial p = q = 0. So, I entered the expression into Wolfram Alpha, which returned the result for q:
q = log ( 3 p ) log ( 4 ) + 2 π i c 1 log ( 4 )
Nothing is said about what is c 1 . Arbitrary constant? If c 1 0 the result could be complex-valued I guess. I tried some calculations using the formula but the results do not make sense.

Answer & Explanation

Carmelo Peck

Carmelo Peck

Beginner2022-08-31Added 12 answers

Other's have discussed the output given by Wolfram|Alpha, so I'll provide only a hint for a solution.
Hint: For every p N with p 1, we have that 4 p is even but 3 p is odd. A natural number cannot be both even and odd, so...
ellynnauwh

ellynnauwh

Beginner2022-09-01Added 7 answers

So to find all solutions, real and complex consider the definition of an exponential. a b = e b log a . Also helps to mention that the complex logarithm of a complex number r e i t is log r + 2 n π i with n an integer. It's to do with the fact that the logarithm function is the inverse of a function that isn't one to one so you've to choose a 'branch'. Anyway...
Now write as e p log 3 = e 4 log q
Take the complex log of both sides so that
p log 3 + 2 n π i = q log 4 where n is an integer and then divide across

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