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Let's analyze each of the given statements to determine which four are true.

1. **Statement 1:** \(\ln e = 1\)  
   This is true because the natural logarithm of \(e\) (Euler's number) is 1. In mathematical terms, \(\ln(e) = 1\) because \(e^1 = e\).

2. **Statement 2:** \(\ln(e^x) \) and \(e^{\ln x}\) both equal \(x\).  
   This statement is true.  
   - \(\ln(e^x) = x\) because the logarithm function is the inverse of the exponential function.  
   - Similarly, \(e^{\ln x} = x\), for \(x > 0\), because \(e\) raised to the natural logarithm of \(x\) equals \(x\).

3. **Statement 3:** \(\ln e = 10\)  
   This is false. As stated earlier, \(\ln e = 1\), not 10.

4. **Statement 4:** \(\ln(x) = \log_{10}(x)\)  
   This is false. \(\ln(x)\) is the natural logarithm (base \(e\)), while \(\log_{10}(x)\) is the common logarithm (base 10). They are not the same unless \(x = 10\).

5. **Statement 5:** \(\ln(x) = \log_e(x)\)  
   This is true. \(\ln(x)\) is just another way to write \(\log_e(x)\), the logarithm of \(x\) with base \(e\).

6. **Statement 6:** \(\ln(e^x) = e^{\ln x}\) are not equivalent.  
   This is true. While both expressions involve \(e\) and \(\ln\), they are not the same:  
   - \(\ln(e^x) = x\)  
   - \(e^{\ln x} = x\) (when \(x > 0\))

7. **Statement 7:** \(y = e^x\) and \(y = \ln x\) are inverses.  
   This is true. The exponential function \(y = e^x\) and the natural logarithm \(y = \ln x\) are indeed inverse functions of each other.

### Conclusion:
The four true statements are:
1. \(\ln e = 1\)
2. \(\ln(e^x) \) and \(e^{\ln x}\) both equal \(x\)
5. \(\ln(x) = \log_e(x)\)
7. \(y = e^x\) and \(y = \ln x\) are inverses

Would you like more details on any of these statements or have any questions?

### Related Questions:
1. What is the difference between natural logarithms and common logarithms?
2. How do exponential and logarithmic functions relate to each other?
3. What is the domain and range of the natural logarithm function?
4. Can logarithms have negative bases? Why or why not?
5. What are some real-world applications of logarithmic functions?

**Tip:** Remember that the natural logarithm \(\ln(x)\) is always based on Euler's number \(e \approx 2.718\).

Which four statements about natural logarithms are true?

Determine which four statements about natural logarithms (ln) are true, including properties of logarithms and their relationship with exponential functions. Suitable for high school students studying logarithms in Grades 10-12.

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