The expression you've uploaded is:

$\left( 1 - e^{-\frac{8 \, \text{s}}{(100,000 \, \Omega)(20 \times 10^{-6} \, \text{F})}} \right)$

This is an exponential decay formula, commonly seen in RC (resistor-capacitor) circuits, where:

• $R = 100,000 \, \Omega$ (resistance)
• $C = 20 \times 10^{-6} \, \text{F}$ (capacitance)
• $t = 8 \, \text{s}$ (time)

The expression does not involve taking the natural logarithm directly (ln). However, the exponential term uses e, which is the base of the natural logarithm.

Step-by-Step Instructions to Enter Into a Calculator:

1. Calculate the time constant $\tau$: $\tau = R \times C = 100,000 \, \Omega \times 20 \times 10^{-6} \, \text{F} = 2 \, \text{s}$

2. Find the ratio $\frac{t}{\tau}$: $\frac{t}{\tau} = \frac{8 \, \text{s}}{2 \, \text{s}} = 4$

3. Substitute into the exponent: $1 - e^{-4}$

4. Calculate the exponential term $e^{-4}$:
   On most calculators, you can input e^(-4) using an e^x button or similar.

   $e^{-4} \approx 0.0183$

5. Subtract from 1: $1 - 0.0183 = 0.9817$

So, the final result of the expression is approximately 0.9817.

Would you like more details or have any specific questions?

Further Questions:

1. What does the time constant $\tau$ represent in RC circuits?
2. How does the value of $t$ (time) affect the result of the expression?
3. Why does this formula use the exponential function $e$?
4. How is the charging and discharging process of a capacitor modeled by this equation?
5. What is the significance of the factor $1 - e^{-t/\tau}$ in the formula?

Tip:

For such calculations, it's handy to use scientific calculators or online tools that directly allow exponential functions like e^x.