How to solve log_2 x=x−1 I thought to write 2^(x−1)=x What should I do next?

anraszbx

anraszbx

Answered question

2022-11-09

How to solve log 2 x = x 1
y = log 2 x = x 1
I thought to write
2 ( x 1 ) = x
What should I do next?

Answer & Explanation

Tasinazzokbc

Tasinazzokbc

Beginner2022-11-10Added 17 answers

Using Derivative Test
Given log 2 ( x ) = x 1 , Here x > 0
As you write here 2 x 1 = x 2 x = 2 x
Now Here we have to find roots of f ( x ) = 0 , Where f ( x ) = 2 x 2 x
Now f ( x ) = 2 x ln ( 2 ) 2 and f ( x ) = 2 x ( ln 2 ) 2 > 0 x R
So f ( x ) = 0 has no real roots
So Using L M V T , We get f ( x ) = 0 has at most one real roots
and f ( x ) = 0 has at most 2 real roots
So here we get x = 1 and x = 2 are the roots of f ( x ) = 0
pin1ta4r3k7b

pin1ta4r3k7b

Beginner2022-11-11Added 3 answers

Well, log 2 ( x ) is a concave function on R + and
log 2 ( x ) = x 1
clearly holds for x = 1 and x = 2. By concavity, it cannot hold for any other x R +

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