Solving Exponential Equations of the Form e^{x^2} - c d

Exponential equations such as e^{x^2} - c d can appear daunting, but they can be effectively solved using fundamental principles of logarithms and algebra. This article will guide you through the process of finding the value of x in equations of the form e^{x^2} - c d, providing detailed steps for solving such problems.

Example 1: e^{x^2} - 2 4

Let's start with the equation:

e^{x^2} - 2 4 Add 2 to both sides of the equation to isolate e^{x^2}: e^{x^2} 6 Take the natural logarithm (ln) of both sides of the equation to solve for x^2: ln(e^{x^2}) ln(6) x^2 ln(6) Take the square root of both sides to find the value of x: x ±√(ln(6)) By approximation, x ≈ ±1.3385662

Example 2: e^{x^2} - 2 4

This is another instance of the same equation. Here is a step-by-step guide:

e^{x^2} - 2 4 Isolate e^{x^2} by adding 2 to both sides: e^{x^2} 6 Take the natural logarithm (ln) of both sides to solve for x: ln(e^{x^2}) ln(6) x^2 ln(6) Take the square root of both sides to find the value of x: x ±√(ln(6)) By approximation, x ≈ ±1.3385662

Using Algebraic Methods

Another common approach to solving such equations involves algebraic manipulation:

exp(x^2) - 2 4 Add 2 to both sides to isolate exp(x^2): exp(x^2) 6 Taking the natural logarithm of both sides to solve for x: ln(exp(x^2)) ln(6) x^2 ln(6) Solving for x by taking the square root of both sides: x ±√(ln(6)) By approximation, x ≈ ±1.3385662

Another Example: e^{x^2} - 2 4

This example is quite similar, but let's walk through each step:

e^{x^2} - 2 4 Isolate e^{x^2} by adding 2 to both sides: e^{x^2} 6 Taking the natural logarithm of both sides to solve for x: ln(e^{x^2}) ln(6) x^2 ln(6) Solving for x by taking the square root of both sides: x ±√(ln(6)) By approximation, x ≈ ±1.3385662

Mathematical Solutions Explained

In each of these examples, we have utilized the properties of exponential and logarithmic functions to solve for the variable x. Specifically, we have used the property that the natural logarithm of an exponential expression can be simplified. This method allows us to isolate x and then solve for its value using algebraic and logarithmic principles.

Here are the key steps for solving such equations:

Isolate the exponential term. Take the natural logarithm of both sides to simplify the equation. Solve for the exponent. Take the square root of both sides to find the value of x.

This process ensures that we find the correct values of x that satisfy the original equation. Remember to consider both positive and negative roots, as the square root can yield both a positive and negative result.

Conclusion

In summary, solving equations of the form e^{x^2} - c d involves a combination of algebraic manipulation and logarithmic functions. By isolating the exponential term and taking the natural logarithm, we can effectively solve for the variable x. Whether you are working with a specific problem or exploring general mathematical principles, understanding these steps will be invaluable in solving a wide range of exponential equations.