How to solve the following logarithmic equation: n(n−1)3^n=91854

untchick04tm

untchick04tm

Answered question

2022-01-22

How to solve the following logarithmic equation: n(n1)3n=91854

Answer & Explanation

recoronarrv

recoronarrv

Beginner2022-01-22Added 20 answers

To Solve Questions of this Type
There is no algebraic method to solve equations like this. If the number on the right hand side had been 91853 instead then the answer can not be written in terms of common expressions (trigonometry, logs/exponentials, arithmetic, etc). Generally questions of this type have to be answered numerically - which could include a number of ways: trial and error, computer software, graphing, etc.
To Solve this Specific Question (or ones where the answer is an integer)
Try to factorize 91854. Youll
reinosodairyshm

reinosodairyshm

Beginner2022-01-23Added 36 answers

n<1F<91854
n1F is increasing F91854 has no more then 1 root

RizerMix

RizerMix

Expert2022-01-27Added 656 answers

One way to get an approximation of a solution is to use the Lambert W function if we note that n(n1)(n12)2. Start with n(n1)3n=91854 Taking the square root and dividing by 34, we get (n12)3n129185434 Rewriting 3n12 and multiplying by 12log(3), we get 12log(3)(n12)e12log(3)(n12)12log(3)9185434 Therefore, 12log(3)(n12)(12log(3)9185434) Evaluating, we get n6.99578 If n is supposed to be an integer, a good guess would be n=7. Trying n=7 shows that that is indeed the answer.

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