We need to solve the equation:

\[
6 \cdot 2^x = 1
\]

### Step 1: Isolate \( 2^x \)
Divide both sides of the equation by 6:

\[
2^x = \frac{1}{6}
\]

### Step 2: Apply the logarithm
To solve for \( x \), take the natural logarithm of both sides:

\[
\ln(2^x) = \ln\left(\frac{1}{6}\right)
\]

Using the logarithm property \( \ln(a^b) = b \cdot \ln(a) \):

\[
x \cdot \ln(2) = \ln\left(\frac{1}{6}\right)
\]

### Step 3: Solve for \( x \)
Now solve for \( x \):

\[
x = \frac{\ln\left(\frac{1}{6}\right)}{\ln(2)}
\]

Let’s calculate the value of \( x \).import math

# Calculate x
x_value = math.log(1/6) / math.log(2)
x_valueThe value of \( x \) is approximately \( -2.585 \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How would you solve \( 4 \cdot 3^x = 2 \)?
2. What happens if the base of the exponent is different, such as \( 5^x = 10 \)?
3. How can you express \( x \) using logarithms for \( 7 \cdot 2^x = 14 \)?
4. What if the equation were \( 6 \cdot 2^{2x} = 1 \)? How would you solve for \( x \)?
5. How does the graph of \( y = 6 \cdot 2^x \) look, and how does it help find the solution?

**Tip:** When solving exponential equations, applying logarithms helps you handle the variable in the exponent more easily.

Learn how to solve the exponential equation 6 * 2^x = 1 with detailed steps and calculations. This problem involves exponential equations and logarithms, suitable for advanced high school students.

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