Need help with solving logarithmic equations 2 2 x − 2 x − 6 =...

Riya Hansen

Riya Hansen

Answered

2022-07-02

Need help with solving logarithmic equations 2 2 x 2 x 6 = 0 and 3 2 x 5 3 x + 4 = 0
No clue how to approach this problem..

Answer & Explanation

Caiden Barrett

Caiden Barrett

Expert

2022-07-03Added 20 answers

Hint:
In each case, you have quadratics in b x :
In the first, put y = 2 x . Then you have y 2 y 6 = 0
For the second, put y = 3 x . Then you have y 2 5 y + 4 = 0
Note: Both quadratic equations factor quite nicely.
( 1 ) ( 2 x ) 2 2 x 6 = 0 y = 2 x y 2 y 6 = ( y 3 ) ( y + 2 ) = 0 y = 2 x = 3  or  y = 2 x = 2
We can see immediately that in both solutions, 2 x must necessarily be positive, since 2 > 0. So we're left with
(1) 2 x = 3 ln ( 2 x ) = ln 3 x ln 2 = ln 3 x = ln 3 ln 2
( 2 ) ( 3 x ) 2 5 ( 3 x ) + 4 = 0 y = 3 x y 2 5 y + 4 = ( y 1 ) ( y 4 ) = 0 y = 3 x = 1  or  y = 3 x = 4
So we have
(1) 3 x = 1 ln ( 3 x ) = ln 1 x ln 3 = ln 1 = 0 = 0 x = 0
And we have
(2) 3 x = 4 ln ( 3 x ) = ln 4 = ln ( 2 2 ) = 2 ln 2 x ln 3 = 2 ln 2 x = 2 ln 2 ln 3
Joshua Foley

Joshua Foley

Expert

2022-07-04Added 3 answers

a. Let 2 x = m m 2 m 6 = 0
Solve the quadratic equation to get:
( m + 2 ) ( m 3 ) = 0 m = 2 ,   m = 3
But remember 2 x = m so:
2 x = 2 ,   2 x = 3
ln both sides of the equation to get:
ln 2 x = ln 3 x ln 2 = ln 3 x = ln 3 ln 2
ln 2 x = ln 2
You cannot take the ln of a negative number so that solution is not applicable.
x = ln 3 ln 2
b. Let 3 x = n n 2 5 n + 4 = 0
Solve the quadratic equation to get:
( n 4 ) ( n 1 ) = 0 n = 1 ,   n = 4
But remember 3 x = n so:
3 x = 1 ,   3 x = 4
ln both sides of the equation to get:
ln 3 x = ln 1 x ln 3 = l n 1
but
ln 1 = 0 x = 0
ln 3 x = ln 4 x ln 3 = ln 4 x = ln 4 ln 3
x = 0 ,   x = ln 4 ln 3

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