We are tasked with solving the equation:

$2e^{\frac{x}{5}} - 8 = 0$

Step-by-Step Solution:

1. Isolate the exponential term: $2e^{\frac{x}{5}} = 8$

2. Divide both sides by 2: $e^{\frac{x}{5}} = 4$

3. Take the natural logarithm (ln) of both sides: $\ln\left(e^{\frac{x}{5}}\right) = \ln(4)$

4. Simplify using the logarithmic identity $\ln(e^a) = a$: $\frac{x}{5} = \ln(4)$

5. Solve for $x$ by multiplying both sides by 5: $x = 5 \ln(4)$

6. Calculate $\ln(4)$ and multiply by 5: $x = 5 \times 1.3863 = 6.9315$

Final Answer:

$x \approx 6.9315$

Let me know if you want more details on this problem, and here are some related questions for further practice:

1. Solve $3e^{\frac{x}{4}} - 9 = 0$.
2. Find the value of $x$ if $5e^{x} = 10$.
3. Solve $e^{2x} - 7 = 0$.
4. Determine $x$ if $4e^{\frac{x}{3}} = 12$.
5. Solve $6e^{x} - 18 = 0$.

Tip: When dealing with exponential equations, using natural logarithms is often the easiest way to isolate and solve for $x$.