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# Pre Calc

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Solve the logarithmic equation for x, as in Example 7. (Enter your answers as a comma-separated list.)

2 log(x) = log(2) + log(4x − 6)

Dec 9, 2019

#1
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Solve for x:
(2 log(x))/log(10) = log(4 x - 6)/log(10) + log(2)/log(10)

Subtract log(4 x - 6)/log(10) + log(2)/log(10) from both sides:
-log(2)/log(10) + (2 log(x))/log(10) - log(4 x - 6)/log(10) = 0

Bring -log(2)/log(10) + (2 log(x))/log(10) - log(4 x - 6)/log(10) together using the common denominator log(10):
-(log(2) - 2 log(x) + log(4 x - 6))/log(10) = 0

Multiply both sides by -log(10):
log(2) - 2 log(x) + log(4 x - 6) = 0

log(2) - 2 log(x) + log(4 x - 6) = log(2) + log(1/x^2) + log(4 x - 6) = log((2 (4 x - 6))/x^2):
log((2 (4 x - 6))/x^2) = 0

Cancel logarithms by taking exp of both sides:
(2 (4 x - 6))/x^2 = 1

Multiply both sides by x^2:
2 (4 x - 6) = x^2

Expand out terms of the left hand side:
8 x - 12 = x^2

Subtract x^2 from both sides:
-x^2 + 8 x - 12 = 0

The left hand side factors into a product with three terms:
-(x - 6) (x - 2) = 0

Multiply both sides by -1:
(x - 6) (x - 2) = 0

Split into two equations:
x - 6 = 0 or x - 2 = 0

x = 6 or x - 2 = 0

x = 6     or     x = 2

Dec 9, 2019
#2
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By log properties.....We can write

log x^2   =  log ( 2 * (4x - 6)  )

log x^2 =  log  [ 8x - 12 ]          we can solve this

x^2  =  8x  - 12

x^2  - 8x  + 12  = 0       factor

(x - 6) ( x - 2)  =  0

Set  each factor to 0  and solve for x  and we get that   x  = 6   and  x  = 2

Dec 9, 2019