We need to solve the equation:

\[
100\left(1 + \frac{0.0044}{3}\right)^{4x} = 101
\]

Let's solve it step by step.

1. **Divide both sides by 100:**

\[
\left(1 + \frac{0.0044}{3}\right)^{4x} = \frac{101}{100}
\]

2. **Simplify the term inside the parentheses:**

\[
1 + \frac{0.0044}{3} = 1 + 0.00146667 = 1.00146667
\]

So the equation now becomes:

\[
(1.00146667)^{4x} = 1.01
\]

3. **Take the natural logarithm (ln) of both sides to bring down the exponent:**

\[
\ln\left((1.00146667)^{4x}\right) = \ln(1.01)
\]

Using the logarithm power rule:

\[
4x \cdot \ln(1.00146667) = \ln(1.01)
\]

4. **Solve for \(x\):**

\[
x = \frac{\ln(1.01)}{4 \cdot \ln(1.00146667)}
\]

Now, let's calculate the value of \(x\).

\[
x \approx \frac{\ln(1.01)}{4 \cdot \ln(1.00146667)} \approx \frac{0.00995033}{4 \cdot 0.00146652} \approx \frac{0.00995033}{0.00586608} \approx 1.696
\]

So, \(x \approx 1.696\).

Would you like further details or explanations?

Here are some related questions:

1. How do logarithm properties simplify equations?
2. What is the difference between natural logarithms and common logarithms?
3. How can you solve exponential equations without using logarithms?
4. What are real-life applications of this type of exponential equation?
5. How does the base of an exponential function affect its graph?

**Tip:** Always check your final answer by substituting it back into the original equation to verify correctness.

Learn how to solve the exponential equation 100(1 + 0.0044/3)^(4x) = 101 with a detailed step-by-step solution. This problem involves exponential equations and logarithmic properties. Suitable for advanced high school students.

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