Determine all real values of x such that: log_(2)(2^(x-1) + 3^(x+1)) = 2x - log_(2)(3^x)

bergvolk0k

bergvolk0k

Answered question

2022-10-23

Find all values of x
log 2 ( 2 x 1 + 3 x + 1 ) = 2 x log 2 ( 3 x )
Let u = 2 x and let y = 3 x
For ease, let log 2 be represented by just log so:
Then, log ( u / 2 + 3 y ) = log ( u 2 ) log ( y ), which means, log ( u / 2 + 3 y ) = log ( u 2 / y ) and so:
u / 2 + 3 y = u 2 / y u y 2 + 3 y 2 u 2 = 0
Factoring a little, y ( u / 4 + 3 y ) + u ( y / 4 u ) = 0 Doesn't help, replace back:
2 x 1 3 x + 3 2 x + 1 2 2 x = 0. But this is all I can go upto? A hint?

Answer & Explanation

Besagnoe9

Besagnoe9

Beginner2022-10-24Added 9 answers

Hint: Your last equation is simply ( u 2 y ) ( u + 1.5 y ) = 0. Since u , y > 0, we get u = 2 y. See my comments for more information.
Lara Cortez

Lara Cortez

Beginner2022-10-25Added 3 answers

the main idea is to write 2 x as
log 2 2 2 x

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