Solving the Equation e^{3x}+2=1 for x

In this article, we will explore the process of solving the exponential equation e^{3x}+2=1 to find the value of x. This type of equation is commonly encountered in high school mathematics, particularly in the study of algebra and exponential functions.

Step 1: Isolate the Exponential Term

The first step in solving this equation is to isolate the exponential term e^{3x} on one side of the equation. We can do this by subtracting 2 from both sides of the equation:

e^{3x}+2=1 e^{3x}=1-2 e^{3x}=-1

Step 2: Take the Natural Logarithm of Both Sides

Now that we have the exponential term isolated, we can take the natural logarithm of both sides of the equation. The natural logarithm function, denoted as ln, is the inverse of the exponential function e^x. This allows us to "undo" the exponential function and solve for x.

ln(e^{3x})=ln(-1) 3x=ln(-1)

Step 3: Solve for x

The final step is to solve for x by dividing both sides of the equation by 3:

3x=ln(-1) x=ln(-1)/3

However, there is a problem here. The natural logarithm of a negative number is not defined in the real number system. This means that there is no real value of x that satisfies the original equation e^{3x}+2=1.

In conclusion, the equation e^{3x}+2=1 has no real solution for x. This is because the natural logarithm of a negative number is not defined in the real number system, and there is no real value of x that satisfies the equation.