Find the exact value of x for the equation (3^x)(4^(2x+1))=6^(x+2)

Aleah Avery

Aleah Avery

Answered question

2022-11-09

Solving 3 x 4 2 x + 1 = 6 x + 2
Find the exact value of x for the equation ( 3 x ) ( 4 2 x + 1 ) = 6 x + 2
Give your answer in the form ln a ln b where a and b are integers.
I have tried using a substitution method, i.e. putting 2 2 to be y, but I have ended up with a complicated equation that hasn't put me any closer to the solution in correct form. Any hints or guidance are much appreciated.
Thanks

Answer & Explanation

naudiliwnw

naudiliwnw

Beginner2022-11-10Added 12 answers

( 3 x ) ( 4 2 x + 1 ) = 6 x + 2
3 x 2 2 ( 2 x + 1 ) = 2 x + 2 3 x + 2
2 2 ( 2 x + 1 ) 3 x = 2 x + 2 3 x + 2
2 2 ( 2 x + 1 ) 2 x + 2 = 3 x + 2 3 x
2 2 ( 2 x + 1 ) ( x + 2 ) = 3 ( x + 2 ) x
2 3 x = 3 2
8 x = 9
Taking natural logarithm on both sides, we get
ln 8 x = ln 9
x = ln 9 ln 8
To be a bit more precise i.e. simplifying a bit more we can have,
x = 2 ln 3 3 ln 2
InjegoIrrenia1mk

InjegoIrrenia1mk

Beginner2022-11-11Added 3 answers

Take the logarithm of both sides, you will end up with a first degree equation:
x ( ln 3 + 2 ln 4 ln 6 ) = ln 4 + 2 ln 6
x ln 8 = ln 9...
Edit: computational error fixed

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