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

Lorena Beard

Lorena Beard

Answered question

2022-07-01

How to solve q = ln n ln b + ln q + ln ln n

Answer & Explanation

Janiyah Patton

Janiyah Patton

Beginner2022-07-02Added 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 ]

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?