solve for the x in the equation 1/2 log2 81 +log2(x^2- x/3) = 1

To solve for x in the equation 1/2 log2 81 + log2(x^2 - x/3) = 1, we can follow these steps:

Step 1: Simplify the equation by using the properties of logarithms.

First, we can simplify the expression 1/2 log2 81. Using the property loga b^c = c loga b, we can rewrite it as log2 81^(1/2). Since 81^(1/2) is equal to 9, we have log2 9.

So, the equation becomes log2 9 + log2(x^2 - x/3) = 1.

Step 2: Combine the logarithms using the property loga b + loga c = loga (b * c).

Applying this property to the equation, we have log2 (9 * (x^2 - x/3)) = 1.

Step 3: Simplify the expression inside the logarithm.

Multiplying 9 by (x^2 - x/3), we get 9x^2 - 3x.

So, the equation becomes log2 (9x^2 - 3x) = 1.

Step 4: Rewrite the equation in exponential form.

In exponential form, the equation loga b = c is equivalent to a^c = b.

Using this form, we have 2^1 = 9x^2 - 3x.

Step 5: Simplify and rearrange the equation.

Simplifying 2^1 to 2, we have 2 = 9x^2 - 3x.

Rearranging the equation, we get 9x^2 - 3x - 2 = 0.

Step 6: Solve the quadratic equation.

To solve the quadratic equation 9x^2 - 3x - 2 = 0, we can use factoring, completing the square, or the quadratic formula.

Factoring the equation, we have (3x + 2)(3x - 1) = 0.

Setting each factor equal to zero, we get 3x + 2 = 0 and 3x - 1 = 0.

Solving these equations, we find x = -2/3 and x = 1/3.

So, the solutions to the equation 1/2 log2 81 + log2(x^2 - x/3) = 1 are x = -2/3 and x = 1/3.