Find value of x in a logarithmic equation If 2^((log_(2)3)^x) =3^(( log_3 2)^x)

Tyson Atkins

Tyson Atkins

Answered question

2022-10-30

Find value of x in a logarithmic equation
If
2 ( log 2 3 ) x = 3 ( log 3 2 ) x
then what is the value of x in this equation? Could taking log on both sides help?

Answer & Explanation

periasemdy

periasemdy

Beginner2022-10-31Added 15 answers

Let y = ( log 2 3 ) x and note that ( log 3 2 ) x = 1 y so you are trying to solve:
2 y = 3 1 / y
or
2 y 2 = 3
So y = log 2 3 and x = 1 / 2. (Note: y positive.)
Stephany Wilkins

Stephany Wilkins

Beginner2022-11-01Added 2 answers

note:
log a b = 1 log b a , log a b = log b log a
Taking log of both sides:
( log 2 3 ) x log 2 = ( log 3 2 ) x log 3
log 2 log 3 = ( log 2 3 ) x ( log 2 3 ) x
log 3 2 = ( log 2 3 ) 2 x
( log 2 3 ) 1 = ( log 2 3 ) 2 x
( log 2 3 ) 1 = ( log 2 3 ) 2 x
x = 1 2

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