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
logx(8x-3) - logx 4=2 Note that we can write
logx ([ 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