Question

Consider the exponential function ${\mathbf{e}}^{\mathbf{-}\mathbf{x}}$. Evaluate this function for x = 1, 10,000, and 0.01.

Short answer

The value of exponent function for $x=1,10,000$is $0$ and the value of exponent function for $x=0.01$ is $0.99$.

Step-by-step solution

Identification of given data

The given data can be listed below,

• values of the function are, $x=1,10,000\mathrm{and}0.01$.

Concept/Significance of exponential function

The exponential and logarithmic functions are diametrically opposed. The exponential function is reversed by the logarithmic function.

Determination of values for function x

The function can be written as,

$$\begin{gathered} y=e^{-x} \\ \mathrm{In}y={\mathrm{Ine}}^{-x} \end{gathered}$$

…(i)

Substitute first value of x in the above.

$$\begin{gathered} \mathrm{In}y=\mathrm{In}{e}^{-110000} \\ =110000 \\ 2.3030\log y=-110000 \\ \log y=-47763.8 \\ y={10}^{-47763.8} \\ =0 \end{gathered}$$

Thus, the value of exponent function for $x=1,10,000$is $0$.

Substitute second value of x in equation (i),

$$\begin{gathered} \mathrm{In}y=\mathrm{In}{e}^{-0.01} \\ =-0.01 \\ 2.303\log y=-0.01 \\ \log y=-0.004342 \\ y={10}^{-0.004342} \\ =0.99 \end{gathered}$$

Thus, the value of exponent function for $x=0.01$is $0.99$.