To solve the equation \(\ln(0.1x) = 0.4\), follow these steps:

### Step 1: Rewrite the equation
\[
\ln(0.1x) = 0.4
\]

### Step 2: Exponentiate both sides
Recall that if \(\ln(a) = b\), then \(a = e^b\). So, exponentiating both sides gives:
\[
0.1x = e^{0.4}
\]

### Step 3: Solve for \(x\)
Now, divide both sides of the equation by 0.1 to isolate \(x\):
\[
x = \frac{e^{0.4}}{0.1}
\]

### Step 4: Calculate the value
Use the approximation \(e^{0.4} \approx 1.4918\):
\[
x \approx \frac{1.4918}{0.1} = 14.918
\]

Thus, the solution is approximately:
\[
x \approx 14.918
\]

Would you like further details or clarification on any step?

Here are 5 related questions:
1. How do you solve logarithmic equations with different bases?
2. What are the properties of natural logarithms?
3. How can you verify the solution of a logarithmic equation?
4. What is the relationship between exponential and logarithmic functions?
5. How does the change of base formula for logarithms work?

**Tip:** Always check your solution by substituting it back into the original equation to ensure it satisfies the equation.

Learn how to solve the logarithmic equation ln(0.1x) = 0.4 with a detailed step-by-step explanation. This solution covers logarithmic properties and the relationship between logarithmic and exponential functions. Suitable for students in Grades 10-12.

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