# Solve for x log base x of 1/27=-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 .
Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.
Solve the equation for .
Rewrite the equation as .
Multiply by .
Multiply by .
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 .
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 1/27=-3