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.