"How to solve a Non-algebraic equation? Whilst working with exponential boom and decay, I encountered a hassle wherein I need to solve an equation related to logarithm. I could not separate or could not make it explicit. y=ln((1+r)(1−delta))/ln((1+ry)(1−delta)). where r and delta are positive, both are smaller than 1. I need the solution of this in the form of; y=f(r,delta). I tried numerical solution also but could not find a best way. Any hint or help is appreciated. "

Logan Knox

Logan Knox

Answered question

2022-09-22

How to solve a Non-algebraic equation?
Whilst working with exponential boom and decay, I encountered a hassle wherein I need to solve an equation related to logarithm. I could not separate or could not make it explicit.
y = ln ( ( 1 + r ) ( 1 δ ) ) ln ( ( 1 + r y ) ( 1 δ ) ) . where r and δ are positive, both are smaller than 1.
I need the solution of this in the form of; y = f ( r , δ ). I tried numerical solution also but could not find a best way. Any hint or help is appreciated.

Answer & Explanation

Lilliana Mason

Lilliana Mason

Beginner2022-09-23Added 11 answers

y = log ( ( 1 + r ) ( 1 δ ) ) log ( ( 1 + r y ) ( 1 δ ) )
For the time being, let
a = log ( ( 1 + r ) ( 1 δ ) ) and b = log ( 1 δ )
which make
y = a log ( 1 + r y ) + b 1 y = log ( 1 + r y ) + b a
Now, some basic manipulations
( 1 + r y ) = ( 1 + b r a ) + r a log ( 1 + r y )
Let
c = 1 + b r a and d = r a and x = 1 + r y
which make
x = c + d log ( x ) x = d W ( 1 d e c d )
Go back to y and replace ( a , b , c , d ) by their definitions in terms of ( r , δ )

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