How can I solve this nature log equation? ln(x+2)=e^((x-4)) Is there any way to solve this equation without graphing or using GDC ?

omvamen71

omvamen71

Answered question

2022-09-03

How can I solve this nature log equation?
l n ( x + 2 ) = e ( x 4 )
Is there any way to solve this equation without graphing or using GDC ?
Thank you

Answer & Explanation

Quinn Alvarez

Quinn Alvarez

Beginner2022-09-04Added 13 answers

The presence of two layers of exponentials does not give any hope for analytical solutions and numerical methods are required.
Consider
f ( x ) = e x 4 log ( x + 2 )
f ( x ) = e x 4 1 x + 2
f ( x ) = e x 4 + 1 ( x + 2 ) 2
The first derivative cancels for
x = W ( e 6 ) 2 2.49666
( W ( z ) being Lambert function); at this point
f ( x ) = 1 W ( e 6 ) + W ( e 6 ) 6 1.28095 < 0
On the other side, by inspection, f ( 1 ) = 1 e 5 is a small positive number; the second derivative being always positive, there are two roots on each side of x .
Let us use Newton method.
For the first root, let us select x 0 = 1; the successive iterates will then be
x 1 = 0.993216345093696
x 2 = 0.993192968155096
x 3 = 0.993192967879949
which is the solution for fifteen significant digits.
For the second root, let select x 0 = 5 (twice the value of x ); the successive iterates will then be
x 1 = 4.70009929479913
x 2 = 4.64012704349585
x 3 = 4.63807063896736
x 4 = 4.63806831105211
x 5 = 4.63806831104913
which is the solution for fifteen significant digits.

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