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.