SOLUTION: Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4.

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Question 1209850: Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4.
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
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Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4.
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You are given  

    log%28%284n%29%2C+%2840%2Asqrt%283%29%29%29 = log%28%282n%29%2C%2810%29%29.    (1)


Let x be the common numerical value of each side of (1), so we can write

    log%28%284n%29%2C+%2840%2Asqrt%283%29%29%29 = x,    (2)

    log%28%282n%29%2C%2810%29%29    = x.    (3)


Then we can rewrite (2) and (3) in this form, respectively

    %284n%29%5Ex = 40%2Asqrt%283%29,        (2')

    %282n%29%5Ex = 10.            (3')


Divide equation (2') by equation (3').  You will get

    %284n%29%5Ex%2F%282n%29%5Ex = 4%2Asqrt%283%29.


You can simplify left side

    2%5Ex = 4%2Asqrt%283%29.


Hence,

    x = log%282%2C%284%2Asqrt%283%29%29%29.


        At this point, we solved half of the problem 
        and have found the numerical value of each 
        expression in the left side of equations (2) and (3).
        Now we are going to make next step and to find n and n%5E4.


We can write equation (3) in the form

    log%28%282n%29%2C%2810%29%29    =  log%282%2C%284%2Asqrt%283%29%29%29.    (4)


Let y be the common numerical value of each side of (4), so we can write

    log%28%282n%29%2C%2810%29%29 = y,                  (5)

    log%282%2C%284%2Asqrt%283%29%29%29 = y.                   (6)


Then we can rewrite (5) and (6) in this form, respectively

    %282n%29%5Ey = 10,                       (5')

    2%5Ey = 4%2Asqrt%283%29.                      (6')


Divide equation (5') by equation (6').  You will get

    n%5Ey = 10%2F%284%2Asqrt%283%29%29 = 5%2F%282%2Asqrt%283%29%29 = %285%2Asqrt%283%29%29%2F6.


Take logarithm of both sides

    y*log(n) = log%28%28%285%2Asqrt%283%29%29%2F6%29%29%29.    (7)


In this formula, it does not matter, which base of logarithm I use.  
Let's the base of logarithm in this equation be 10.


From equation (7), 

    log(n) = %281%2Fy%29%2Alog%28%28%285%2Asqrt%283%29%29%2F6%29%29.


We can substitute here expression (6) for y.  We get then

    log(n) = 1%2Flog%282%2C%284%2Asqrt%283%29%29%29 * log%28%28%285%2Asqrt%283%29%29%2F6%29%29.


This is the  "precise"  expression for log(n).
We can find its numerical value 

    log(n) = 0.159379381%2F2.79248125 = 0.057074.


Hence,  n = 10%5E0.057074 = 1.140444,  and  n%5E4 = 1.140444%5E4 = 1.691601, approximately.



At this point, the solution is complete and the value of  n%5E4  is found.


ANSWER.  n%5E4 = 1.140444%5E4 = 1.691601,  approximately.

Solved.

After completing this solution, I checked equation (1)
and found that both sides are equal at n = 1.140444.