To determine the number of points of intersection for the graphs of the equations [tex]\( y = e^x \)[/tex] and [tex]\( xy = 20 \)[/tex], we need to solve the system of equations simultaneously.
1. First, let's write down the given equations:
[tex]\[
y = e^x
\][/tex]
[tex]\[
xy = 20
\][/tex]
2. Substitute [tex]\( y = e^x \)[/tex] from the first equation into the second equation:
[tex]\[
x \cdot e^x = 20
\][/tex]
3. This equation [tex]\( x \cdot e^x = 20 \)[/tex] is a transcendental equation and doesn't lend itself to simple algebraic manipulation. However, we can solve it using properties of exponentials and the Lambert W function where appropriate.
4. Recall that the Lambert W function [tex]\( W(z) \)[/tex] is defined as the inverse of the function [tex]\( z = W(z)e^{W(z)} \)[/tex]. This means [tex]\( W(ze^z) = z \)[/tex].
5. Rewrite [tex]\( x \cdot e^x = 20 \)[/tex] in a form suitable for the Lambert W function:
[tex]\[
x \cdot e^x = 20
\][/tex]
We can rewrite in the form [tex]\( ze^z = C \)[/tex] where [tex]\( C \)[/tex] is a constant:
[tex]\[
x e^x = 20 \implies (x e^x) = 20
\][/tex]
6. Let [tex]\( u = x \)[/tex]; therefore, the equation transforms into:
[tex]\[
u e^u = 20
\][/tex]
Thus, [tex]\( u = W(20) \)[/tex], where [tex]\( W \)[/tex] is the Lambert W function.
7. The Lambert W function [tex]\( W(z) \)[/tex] can have multiple values (branches), specifically [tex]\( W_0(z) \)[/tex] and [tex]\( W_{-1}(z) \)[/tex], since the Lambert W function is multi-valued for certain ranges of [tex]\( z \)[/tex]. Here, [tex]\( W_0 \)[/tex] denotes the principal branch and [tex]\( W_{-1} \)[/tex] denotes the lower branch.
8. Therefore, the solutions are:
[tex]\[
x = W_0(20) \quad \text{and} \quad x = W_{-1}(20)
\][/tex]
9. Since the Lambert W function can have up to two real values for positive arguments, there are two possible solutions for [tex]\( x \)[/tex]:
[tex]\[
x_1 = W_0(20)
\][/tex]
[tex]\[
x_2 = W_{-1}(20)
\][/tex]
10. Corresponding to these two values of [tex]\( x \)[/tex], we will have two values of [tex]\( y \)[/tex]:
[tex]\[
y_1 = e^{x_1} = e^{W_0(20)}
\][/tex]
[tex]\[
y_2 = e^{x_2} = e^{W_{-1}(20)}
\][/tex]
Thus, the total number of points of intersection for the graphs of the equations is:
[tex]\[
\boxed{2}
\][/tex]
