How to solve q = ln &#x2061;<!-- ⁡ --> <mrow class="MJX-TeXAto

Lorena Beard 2022-07-01 Answered
How to solve q = ln n ln b + ln q + ln ln n
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Answers (1)

Janiyah Patton
Answered 2022-07-02 Author has 12 answers
Let q = e x so that ln q = x .. Also, put A = ln n ,, B = ln b ,, and C = ln ln n .. Then your equation takes the form
e x = A B + x + C
Multiplying both sides by B + x + C gives
e x ( B + x + C ) = A
Now make the variable change u = B + x + C ,, so that x = u B C ..Then we get
e u B C u = A
Multiplying both sides by e B + C gives
e u u = A e B + C
Now we can express the solution in terms of the Lambert function:
u = W ( A e B + C )
Hence, recalling that u = B + x + C and q = e x we get
B + x + C = W ( A e B + C )
x = W ( A e B + C ) B C
e x = exp [ W ( A e B + C ) B C ]
q = exp [ W ( A e B + C ) B C ]
q = exp [ W ( ( ln n ) e ln b + ln ln n ) ln b ln ln n ]
(a few minutes later) Using e ln b + ln ln n = e ln b e ln ln n = b ln n , we get
q = exp [ W ( b ( ln n ) 2 ) ln b ln ln n ]
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