Solving Equations of the Form 4^x 4^{1/x} k using Algebra and Logarithms

When faced with equations in the form of 4x 41/x k, it can be challenging to find the solution without resorting to numerical methods. This article explores how these types of problems can be solved using algebraic techniques, specifically through the transformation of the equation into a standard quadratic form, followed by the application of the quadratic formula. We also discuss the role of logarithms in such problems, providing a step-by-step approach to finding the exact value of x.

Introduction to the Problem

The equation we are considering is of the form 4x 41/x k. This type of equation is common in various mathematical and scientific contexts, such as in the study of exponential growth, signal processing, and even in certain optimization problems.

Solution Approach: Algebraic and Quadratic Transformation

Let's consider the specific equation 4x 41/x 18. We can start by letting y 4x. This transformation allows us to rewrite 41/x in terms of y, as follows:

41/x (4x)1/x2 y1/x2

Substituting y for 4x and y1/x2 for 41/x, the original equation becomes:

y y1/x2 18

Now, to eliminate the fractional exponent, we can multiply both sides by y:

y2 4 18y

Rearranging this into a standard quadratic equation:

y2 - 18y 4 0

Applying the quadratic formula, where a 1, b -18, and c 4, we get:

y frac{-b pm sqrt{b^2 - 4ac}}{2a}

y frac{18 pm sqrt{324 - 16}}{2}

y frac{18 pm sqrt{308}}{2}

y frac{18 pm 2sqrt{77}}{2}

y 9 pm sqrt{77}

Since y 4x must be positive, we discard the solution where y 9 - sqrt{77}, as it is less than zero. Thus, we have:

y 9 sqrt{77}

Since y 4x, we can rewrite it as:

4x 9 sqrt{77}

To solve for x, we take the logarithm base 4 of both sides:

x log_{4}(9 sqrt{77})

Alternative Solutions: Logarithmic Approach

Another method involves expressing the equation in terms of logarithms. Starting with the equation:

4x 41/x 18

We can derive the solutions directly by inspection, as shown in the premises:

x frac{1}{2} or x 2

Let's validate these solutions by substituting x 2:

42 41/2 16 sqrt{4} 16 2 18

And for x frac{1}{2}:

4frac{1}{2} 41/(frac{1}{2}) sqrt{4} 42 2 16 18

Generalization of the Solution

To generalize, let's consider the equation 4x 41/x k. The solution can be expressed as:

x log_{4}(9 sqrt{77})

This solution can also be written in terms of natural logarithms or base 10 logarithms using the change of base formula:

x frac{log(9 sqrt{77})}{log(4)} or x frac{ln(9 sqrt{77})}{ln(4)}

Conclusion

The problem of solving equations of the form 4x 41/x k can be approached algebraically by transforming it into a standard quadratic equation and applying the quadratic formula. Alternatively, logarithmic methods provide a direct solution through inspection. Both methods are valid and can be used depending on the specific problem constraints and the desired precision of the solution.