# Solve for x log base x of 8=-3

Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Solve for
Rewrite the expression using the negative exponent rule .
Solve for .
Multiply each term by and simplify.
Multiply each term in by .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Rewrite the equation as .
Move to the left side of the equation by subtracting it from both sides.
Factor the left side of the equation.
Rewrite as .
Rewrite as .
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Simplify.
Apply the product rule to .
Raise to the power of .
Multiply by .
One to any power is one.
Set equal to and solve for .
Set the factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
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 .
Factor out of .
Multiply by .
Multiply by .
The final answer is the combination of both solutions.
The solution is the result of and .
Solve for x log base x of 8=-3