a=b^x+c^x , How to solve for x? If a=b^x, then x could be written in terms of a and b; x=(log(a))/(log(b)) What about a=b^x+c^x?

Raiden Barr

Raiden Barr

Answered question

2022-10-27

a = b x + c x , How to solve for x?
If a = b x , then x could be written in terms of a and b; x = log ( a ) log ( b )
What about a = b x + c x ?
Could x be written in terms of a , b and c? x = ?

Answer & Explanation

Jovanni Salinas

Jovanni Salinas

Beginner2022-10-28Added 18 answers

The best we can do (in terms of algebraic solutions) is as follows: rewrite the equation as
a = ( e x ) ln ( b ) + ( e x ) ln ( c )
setting y = e x , this becomes an equation on y:
a = y ln ( b ) + y ln ( c )
Or, rearranging,
y ln ( b ) + y ln ( c ) a = 0
In other words, we're looking for x > 0 that satisfy
x β + x γ a = 0
There is no general way to solve for x if β , γ are integers, let alone if β and γ can be arbitrary real numbers.

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