To solve the equation \( (1.056)^x = 100 \), we follow these steps:

### Step 1: Take the natural logarithm (ln) of both sides
\[
\ln((1.056)^x) = \ln(100)
\]

### Step 2: Apply the logarithmic property \( \ln(a^b) = b\ln(a) \)
\[
x \cdot \ln(1.056) = \ln(100)
\]

### Step 3: Isolate \(x\) by dividing both sides by \( \ln(1.056) \)
\[
x = \frac{\ln(100)}{\ln(1.056)}
\]

### Step 4: Calculate the values
\[
\ln(100) \approx 4.6052
\]
\[
\ln(1.056) \approx 0.0545
\]

Now divide:
\[
x \approx \frac{4.6052}{0.0545} \approx 84.5
\]

Thus, \(x \approx 84.5\).

Would you like more details on logarithmic properties or any other questions?

Here are 5 related questions:
1. How can you verify this solution using a calculator?
2. What is the logarithmic function used for in real-life applications?
3. How would the result change if the base were a different number?
4. What does the value of \(x\) represent in a real-world context?
5. Can this problem be solved using any other logarithmic base?

**Tip:** Logarithms help simplify exponential equations, making them easier to solve by turning multiplicative problems into additive ones.

Learn how to solve the exponential equation (1.056)^x = 100 by applying logarithmic properties. This step-by-step solution explains how to use the natural logarithm to find the value of x, making it suitable for students in Grades 9-12.

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