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To solve the equation \( 2^x = 7 \), we can follow these steps:

### Step 1: Apply the logarithm
We apply the natural logarithm (or any logarithm) to both sides of the equation to bring the exponent down:
\[
\ln(2^x) = \ln(7)
\]

### Step 2: Use the logarithm power rule
The logarithmic power rule allows us to bring the exponent \( x \) down:
\[
x \cdot \ln(2) = \ln(7)
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \) by dividing both sides by \( \ln(2) \):
\[
x = \frac{\ln(7)}{\ln(2)}
\]

### Step 4: Calculate the value
Using a calculator:
\[
x = \frac{\ln(7)}{\ln(2)} \approx \frac{1.9459}{0.6931} \approx 2.8074
\]

Rounding to three decimal places:
\[
x \approx 2.807
\]

### Final Answer:
The correct choice is:
- **A.** \( x = 2.807 \)

Would you like further details or clarification on any of the steps?

Here are 5 related questions to expand your understanding:
1. How would you solve \( 3^x = 10 \) using the same logarithmic method?
2. What happens if the base of the exponential equation is less than 1?
3. How can logarithms help in solving equations involving powers of e, such as \( e^x = 5 \)?
4. What is the significance of the natural logarithm \( \ln \) in solving exponential equations?
5. How would you solve an equation like \( 2^{2x} = 16 \)?

**Tip:** When solving exponential equations, applying logarithms helps simplify expressions where the variable is in the exponent.

Solve the following exponential equation: 2^x = 7.

Learn how to solve the exponential equation 2^x = 7 using logarithms. This problem demonstrates key concepts in exponential equations and logarithmic transformations. Suitable for high school students in Grades 9-12.

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