Show that x = 2 ln &#x2061;<!-- ⁡ --> ( 3 x &#x2212;<!-- - --> 2 ) can be wr

Matilda Webb

Matilda Webb

Answered question

2022-05-14

Show that x = 2 ln ( 3 x 2 ) can be written as x = 1 3 ( e x / 2 + 2 )
Is there a rule for this?

Answer & Explanation

Bentyrchjurvk

Bentyrchjurvk

Beginner2022-05-15Added 15 answers

Solve for the "other" x. Notice that:
x = 2 ln ( 3 x 2 ) x 2 = ln ( 3 x 2 ) e x / 2 = 3 x 2 e x / 2 + 2 = 3 x 1 3 ( e x / 2 + 2 ) = x
as desired.
Hailee Stout

Hailee Stout

Beginner2022-05-16Added 7 answers

It’s just the relationship between logs and exponents. Starting with x = 2 ln ( 3 x 2 ), divide both sides by 2 to get
x 2 = ln ( 3 x 2 ) .
Now exponentiate on both sides:
e x 2 = e ln ( 3 x 2 ) = 3 x 2 ,
because by definition e ln u = u for all u > 0. Finally, add 2 to both sides to get
e x 2 + 2 = 3 x ,
and multiply both sides by 1 3 , and you’re done.

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