Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .

Move to the left side of the equation by subtracting it from both sides.

Factor the left side of the equation.

Rewrite as .

Since both terms are perfect cubes, factor using the difference of cubes formula, where and .

Simplify.

Move to the left of .

Raise to the power of .

Set equal to and solve for .

Set the factor equal to .

Add to both sides of the equation.

Set equal to and solve for .

Set the factor equal to .

Use the quadratic formula to find the solutions.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

Move to the left of .

Multiply by .

Simplify .

The final answer is the combination of both solutions.

The solution is the result of and .

Solve for x log base x of 1000=3