#1**0 **

Solve for x:

(log(8 x - 3))/(log(x)) - (log(4))/(log(x)) = 2

Bring (log(8 x - 3))/(log(x)) - (log(4))/(log(x)) together using the common denominator log(x):

(log(8 x - 3) - log(4))/(log(x)) = 2

Multiply both sides by log(x):

log(8 x - 3) - log(4) = 2 log(x)

Subtract 2 log(x) from both sides:

-log(4) - 2 log(x) + log(8 x - 3) = 0

-log(4) - 2 log(x) + log(8 x - 3) = log(1/4) + log(1/x^2) + log(8 x - 3) = log((8 x - 3)/(4 x^2)):

log((8 x - 3)/(4 x^2)) = 0

Cancel logarithms by taking exp of both sides:

(8 x - 3)/(4 x^2) = 1

Multiply both sides by 4 x^2:

8 x - 3 = 4 x^2

Subtract 4 x^2 from both sides:

-4 x^2 + 8 x - 3 = 0

The left hand side factors into a product with three terms:

-(2 x - 3) (2 x - 1) = 0

Multiply both sides by -1:

(2 x - 3) (2 x - 1) = 0

Split into two equations:

2 x - 3 = 0 or 2 x - 1 = 0

Add 3 to both sides:

2 x = 3 or 2 x - 1 = 0

Divide both sides by 2:

x = 3/2 or 2 x - 1 = 0

Add 1 to both sides:

x = 3/2 or 2 x = 1

Divide both sides by 2:

** x = 3/2 or x = 1/2**

Guest Sep 2, 2017

#2**+1 **

log_{x}(8x-3) - log_{x }4=2 Note that we can write

log_{x} ([ 8x - 3] / 4) = 2 In exponential form, we have that

x^2 = ([ 8x - 3] / 4) Multiply both sides by 4

4x^2 = 8x - 3 rearrange as

4x^2 - 8x + 3 = 0 factor as

(2x - 3) (2x - 1) = 0

Setting each factor to 0 and solving for x gives us the possible solutions of x = 3/2 or x = 1/2

Checking x = 3/2 using the change of base method

log ( 9) / log (3/2) - log (4) / log(3/2) = 2

Checking x = 1/2 using the change of base method

log ( 1) / log (1/2) - log (4) / log(1/2) = 2

So both solutions are valid

CPhill Sep 2, 2017