Let's solve the equation:

\[
\ln\left(\frac{5}{10}\right) = \frac{6}{3} \left(\frac{1}{x} - \frac{1}{10}\right)
\]

### Step 1: Simplify both sides of the equation.
1. The natural logarithm expression on the left side can be simplified:
   \[
   \ln\left(\frac{5}{10}\right) = \ln\left(\frac{1}{2}\right)
   \]

2. Simplify the right-hand side:
   \[
   \frac{6}{3} = 2
   \]
   So, the equation becomes:
   \[
   \ln\left(\frac{1}{2}\right) = 2 \left(\frac{1}{x} - \frac{1}{10}\right)
   \]

### Step 2: Solve for \(\frac{1}{x}\).
1. The natural logarithm of \( \frac{1}{2} \) is:
   \[
   \ln\left(\frac{1}{2}\right) = -\ln(2)
   \]
   So the equation becomes:
   \[
   -\ln(2) = 2\left(\frac{1}{x} - \frac{1}{10}\right)
   \]

2. Divide both sides by 2:
   \[
   -\frac{\ln(2)}{2} = \frac{1}{x} - \frac{1}{10}
   \]

3. Add \( \frac{1}{10} \) to both sides to solve for \( \frac{1}{x} \):
   \[
   \frac{1}{x} = -\frac{\ln(2)}{2} + \frac{1}{10}
   \]

### Step 3: Calculate \( x \).
1. Numerically, \(\ln(2) \approx 0.6931\), so:
   \[
   \frac{1}{x} = -\frac{0.6931}{2} + \frac{1}{10} = -0.34655 + 0.1 = -0.24655
   \]

2. Finally, solve for \( x \):
   \[
   x = \frac{1}{-0.24655} \approx -4.06
   \]

Thus, \( x \approx -4.06 \).

Would you like further details or clarification?

### Related Questions:
1. What is the significance of the natural logarithm function in calculus?
2. How do you solve logarithmic equations involving multiple variables?
3. Can you explain properties of logarithms that help simplify equations?
4. How can natural logarithms be used in real-world applications?
5. What are the common mistakes made when solving logarithmic equations?

### Tip:
Always check for extraneous solutions when solving logarithmic equations, as logarithms are only defined for positive inputs.

Learn how to solve the logarithmic equation ln(5/10)=(6/3)((1/x)-(1/10)) step-by-step. This problem involves logarithm properties and equations, suitable for advanced high school students and beyond.

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