To solve the equation \( 2 = \left(1 + \frac{x}{12}\right)^{102} \), we can follow these steps:

### Step 1: Take the natural logarithm of both sides
We apply the natural logarithm (ln) to both sides to simplify the exponent:

\[
\ln(2) = \ln\left(\left(1 + \frac{x}{12}\right)^{102}\right)
\]

Using the logarithmic identity \( \ln(a^b) = b \ln(a) \), this becomes:

\[
\ln(2) = 102 \cdot \ln\left(1 + \frac{x}{12}\right)
\]

### Step 2: Solve for \( \ln\left(1 + \frac{x}{12}\right) \)
Now, divide both sides by 102:

\[
\frac{\ln(2)}{102} = \ln\left(1 + \frac{x}{12}\right)
\]

### Step 3: Exponentiate both sides
To get rid of the logarithm, exponentiate both sides:

\[
e^{\frac{\ln(2)}{102}} = 1 + \frac{x}{12}
\]

Since \( e^{\ln(a)} = a \), this simplifies to:

\[
2^{\frac{1}{102}} = 1 + \frac{x}{12}
\]

### Step 4: Solve for \( x \)
Now subtract 1 from both sides:

\[
2^{\frac{1}{102}} - 1 = \frac{x}{12}
\]

Finally, multiply both sides by 12 to isolate \( x \):

\[
x = 12 \cdot \left(2^{\frac{1}{102}} - 1\right)
\]

### Step 5: Calculate the value
Now let's calculate the numerical value of \( x \):

\[
x = 12 \cdot \left(2^{\frac{1}{102}} - 1\right)
\]

Let me compute this.The solution to the equation \( 2 = \left(1 + \frac{x}{12}\right)^{102} \) is approximately:

\[
x \approx 0.0818
\]

Would you like more details, or do you have any questions?

Here are 5 related questions you can explore:
1. How would the solution change if the exponent 102 were different?
2. Can we solve this equation using logarithms other than the natural log?
3. What does the graph of \( \left(1 + \frac{x}{12}\right)^{102} \) look like?
4. How can this equation be applied in compound interest problems?
5. How does changing the base of the exponential term affect the solution?

**Tip:** When solving equations involving exponents, taking the logarithm is often a useful step to simplify and isolate variables.

This guide provides a step-by-step solution to the exponential equation 2 = (1 + x/12)^102, using logarithms to simplify and solve for x. It covers key mathematical concepts such as logarithmic identities and exponentiation, suitable for high school students in Grades 10-12.

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