Solve this equation for x. ln(x)+ln(x-6)=3

Francisco Proctor

Francisco Proctor

Answered question

2022-07-28

Solve this equation for x.
ln ( x ) + ln ( x 6 ) = 3

Answer & Explanation

ri1men4dp

ri1men4dp

Beginner2022-07-29Added 14 answers

ln ( x ) + ln ( x 6 ) = 3
remember ln ( a ) + ln ( b ) = ln ( a . b )
ln ( e n ) = n ln ( e ) = n
ln ( x . ( x 6 ) ) = ln ( e 3 )
x ( x 6 ) = e 3
x 2 6 x e 3 ) = 0 x 1 , 2 = 6 ± 36 4 e 3 2 = 3 ± 9 e 3 1 = 3 ± 9 e 3
x 1 = 3 9 e 3
x 2 = 3 + 9 e 3
ganolrifv9

ganolrifv9

Beginner2022-07-30Added 4 answers

We must take note that our domain is x>6 due to the ln ( x 6 ).
Using log rules ln ( a ) + ln ( b ) = ln ( a b )
ln ( x ( x 6 ) ) = 3
ln ( x 2 6 x ) = 3
Raising e to both sides.
e ln ( x 2 6 x ) = e 3
Simplifying since e ln ( x 2 6 x ) = x 2 6 x
x 2 6 x = e 3
Recognizing that this is just a quadratic equation and rearranging
x 2 6 x e 3 = 0
Using quadratic formulate
x = 6 ± 36 + 4 e 3 2
x = 6 ± ( 4 ) 9 + e 3 2
x = 3 ± 9 + e 3
Note the 3 9 + e 3 is not valid since it is less than 6, so the only solution is
x = 3 + 9 + e 3

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