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)

mharrigan920 Dec 9, 2019

#1**+1 **

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

Add 6 to both sides:

x = 6 or x - 2 = 0

Add 2 to both sides:

**x = 6 or x = 2**

Guest Dec 9, 2019