To solve this logarithmic equation, we can rewrite it using the property of logarithms that states log_a(b) = c is equivalent to a^c = b.
First, we can rewrite the equation as:
2log1/2x = log1/2(2x^2-x)
Next, we can rewrite the left side of the equation using the power rule of logarithms, which states that n*log_a(b) = log_a(b^n):
log1/2x^2 = log1/2(2x^2-x)
Now, we can remove the logarithms from both sides of the equation by raising 1/2 to the power of each side:
1/2^2 = 2x^2-x
This simplifies to:
1/4 = 2x^2 - x
Rearranging the equation, we get:
2x^2 - x - 1/4 = 0
Now, we can factor this quadratic equation by using the quadratic formula or by standard factorization.
The equation factors to:
(2x - 1/2)(x + 1/2) = 0
Therefore, the solutions are:
2x - 1/2 = 0 OR x + 1/2 = 0
2x = 1/2 OR x = -1/2
x = 1/4 OR x = -1/2
Therefore, the solutions to the equation are x = 1/4 and x = -1/2.
To solve this logarithmic equation, we can rewrite it using the property of logarithms that states log_a(b) = c is equivalent to a^c = b.
First, we can rewrite the equation as:
2log1/2x = log1/2(2x^2-x)
Next, we can rewrite the left side of the equation using the power rule of logarithms, which states that n*log_a(b) = log_a(b^n):
log1/2x^2 = log1/2(2x^2-x)
Now, we can remove the logarithms from both sides of the equation by raising 1/2 to the power of each side:
1/2^2 = 2x^2-x
This simplifies to:
1/4 = 2x^2 - x
Rearranging the equation, we get:
2x^2 - x - 1/4 = 0
Now, we can factor this quadratic equation by using the quadratic formula or by standard factorization.
The equation factors to:
(2x - 1/2)(x + 1/2) = 0
Therefore, the solutions are:
2x - 1/2 = 0 OR x + 1/2 = 0
2x = 1/2 OR x = -1/2
x = 1/4 OR x = -1/2
Therefore, the solutions to the equation are x = 1/4 and x = -1/2.