Solve for 𝑥 in the following expression. Enter your answer in decimal form, rounded to four decimal places. If an answer does not exist, enter DNE.

2𝑒^(𝑥/5)−8=0

𝑥 =

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 \( x \).

Solve for x in the following expression. Enter your answer in decimal form, rounded to four decimal places. If an answer does not exist, enter DNE.

2e^(x/5)−8=0

x =

Learn how to solve the exponential equation 2e^(x/5)−8 = 0 step by step, involving logarithms and exponential functions. This problem highlights key concepts of algebra and is suitable for students in Grades 10-12.

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