To solve the equation \( 2^{x+1} = 6 \), follow these steps:

### Step 1: Isolate the exponential expression
The equation is already in a form where the exponential expression \( 2^{x+1} \) is isolated.

### Step 2: Take the logarithm of both sides
To solve for \( x \), take the logarithm (typically the natural logarithm, \( \ln \)) of both sides:
\[
\ln(2^{x+1}) = \ln(6)
\]

### Step 3: Use the logarithm power rule
Apply the logarithm power rule, which states \( \ln(a^b) = b\ln(a) \):
\[
(x + 1) \ln(2) = \ln(6)
\]

### Step 4: Solve for \( x \)
Now, solve for \( x \) by dividing both sides by \( \ln(2) \) and then subtracting 1:
\[
x + 1 = \frac{\ln(6)}{\ln(2)}
\]
\[
x = \frac{\ln(6)}{\ln(2)} - 1
\]

### Step 5: Simplify the expression (optional)
You can leave the answer in this logarithmic form, or you can approximate it numerically. Let's calculate the approximate value:
\[
x \approx \frac{\ln(6)}{\ln(2)} - 1 \approx \frac{1.791759}{0.693147} - 1 \approx 2.58496 - 1 \approx 1.58496
\]

Thus, the solution is approximately \( x \approx 1.585 \).

Would you like to explore further details or need any more explanations? Here are some related questions you might consider:

1. What are logarithms, and how are they used to solve exponential equations?
2. How can you solve exponential equations without a calculator?
3. What is the difference between natural logarithms and common logarithms?
4. How do logarithmic functions relate to exponential functions?
5. What are some real-world applications of solving exponential equations?
6. How can you check if your logarithmic solution is correct?
7. What is the significance of the base in an exponential equation?
8. How would the solution change if the base of the exponent were different?

**Tip:** When solving logarithmic and exponential equations, it's helpful to remember key logarithm properties, such as the power rule, product rule, and change of base formula.

Learn how to solve the exponential equation 2^(x+1) = 6 with a detailed step-by-step solution. This problem covers exponential equations and logarithms, focusing on the logarithm power rule. Suitable for high school students (Grades 10-12).

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