Let's examine the given expression and prove if the equality holds:

We are asked to verify:

$\ln\left(\frac{1+x}{1-x}\right) = \ln(1+x) - \ln(1-x)$

Step 1: Use logarithmic identity

Recall the logarithmic identity:

$\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$

So, on the right-hand side of the equation, we can apply this identity:

$\ln(1+x) - \ln(1-x) = \ln\left(\frac{1+x}{1-x}\right)$

Step 2: Conclusion

This shows that the right-hand side simplifies to the left-hand side of the equation. Hence, the given equation is true.

$\ln\left(\frac{1+x}{1-x}\right) = \ln(1+x) - \ln(1-x)$

Do you need further clarification on this, or would you like help with another problem? Here are some related questions you might find useful:

1. What are other logarithmic identities that can simplify expressions like this one?
2. How can you solve equations involving logarithms?
3. What are the properties of logarithms and how can they be applied in integration?
4. How do you solve for $x$ in logarithmic equations?
5. What is the graph of the function $\ln\left(\frac{1+x}{1-x}\right)$?

Tip: When dealing with logarithmic expressions, always check if you can apply common identities, like $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$, to simplify the expression before solving.