Final answer:
To solve the equation log2 (2x^3 - 8) - 2log2 x=log2x, simplify the expression, combine the logarithms, set the expressions equal to each other, eliminate the fraction, and solve for x. The solution is x = 2.
Step-by-step explanation:
To solve the equation log2 (2x^3 - 8) - 2log2 x=log2x, we can simplify it step by step.
Apply the property of logarithms which states that the logarithm of a division is equal to the difference of the logarithms of the numerator and denominator. We have log2 (2x^3 - 8) - log2 x^2 = log2 x.
Combine the logarithms on the left side using the property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the factors.
We now have log2 [(2x^3 - 8)/x^2] = log2 x.
Since the bases of the logarithms are the same, we can eliminate the logarithms and set the expressions inside them equal to each other. We get (2x^3 - 8)/x^2 = x.
Multiply both sides of the equation by x^2 to eliminate the fraction. We obtain 2x^3 - 8 = x^3.
Simplify and solve for x. x^3 = 8, so x = 2.