To solve for the expression [tex]\( e^{2x} \)[/tex], we start by identifying the components and understanding the base of the expression.
1. Base [tex]\( e \)[/tex]: The base of the natural exponential function is [tex]\( e \)[/tex], which is an irrational constant approximately equal to 2.71828. The exponential function [tex]\( e^x \)[/tex] represents the function where [tex]\( e \)[/tex] is raised to the power of [tex]\( x \)[/tex].
2. Exponent [tex]\( 2x \)[/tex]: The exponent in the given expression is [tex]\( 2x \)[/tex]. This means that [tex]\( x \)[/tex] is scaled by a factor of 2 before applying the exponential function.
3. Combination: When we combine these, [tex]\( e^{2x} \)[/tex] is interpreted as [tex]\( e \)[/tex] raised to the power of [tex]\( 2x \)[/tex].
So, the expression [tex]\( e^{2x} \)[/tex] represents the exponentiation of the constant [tex]\( e \)[/tex] to the power of twice the variable [tex]\( x \)[/tex].
To summarize:
- Understand the function: The expression is [tex]\( e \)[/tex] raised to the power [tex]\( 2x \)[/tex].
- Variables: [tex]\( x \)[/tex] is the variable in the expression.
- Interpret the exponent: The exponent is [tex]\( 2x \)[/tex], which means you multiply the variable [tex]\( x \)[/tex] by 2 before raising [tex]\( e \)[/tex] to this product.
Thus, the final mathematical expression is [tex]\( e^{2x} \)[/tex].
