Write the expression as a sum and/or difference of logarithms. 

Provide your answer below:

$\ln\left(x\sqrt{1+x^2}\right),\ x>0\ \ =\ $ 


We are given the expression:

\[
\ln\left(x\sqrt{1+x^2}\right)
\]

### Step-by-step solution:
1. **Break the product inside the logarithm**:
   Use the logarithmic property:
   \[
   \ln(a \cdot b) = \ln(a) + \ln(b)
   \]
   Here, $a = x$ and $b = \sqrt{1+x^2}$, so:
   \[
   \ln\left(x\sqrt{1+x^2}\right) = \ln(x) + \ln\left(\sqrt{1+x^2}\right)
   \]

2. **Simplify the square root term**:
   Next, simplify the second logarithmic term. The square root can be written as an exponent:
   \[
   \sqrt{1+x^2} = (1+x^2)^{1/2}
   \]
   Now, apply the logarithmic property:
   \[
   \ln(a^n) = n\ln(a)
   \]
   So:
   \[
   \ln\left(\sqrt{1+x^2}\right) = \frac{1}{2} \ln(1+x^2)
   \]

3. **Final expression**:
   Now combine everything:
   \[
   \ln\left(x\sqrt{1+x^2}\right) = \ln(x) + \frac{1}{2} \ln(1+x^2)
   \]

Thus, the expression written as a sum of logarithms is:

\[
\ln(x) + \frac{1}{2} \ln(1+x^2)
\]

---

Do you have any questions or would you like further details on any part?

Here are 5 related questions:
1. How would you simplify $\ln\left(\frac{a}{b}\right)$?
2. What is the logarithmic property for a power function like $\ln(a^n)$?
3. How can you expand $\ln(ab^2)$ using logarithmic properties?
4. What happens to the logarithmic properties if $x$ is negative?
5. How do we differentiate $\ln\left(x^2 + 1\right)$?

**Tip**: Always remember that logarithmic properties allow you to simplify products, quotients, and powers inside logarithms!

Write the expression as a sum and/or difference of logarithms: $\ln\left(x\sqrt{1+x^2}\right),\ x>0$.

Learn how to expand the logarithmic expression $\ln\left(x\sqrt{1+x^2}\right)$ using properties of logarithms. This step-by-step solution breaks down the product and simplifies the square root to express the result as a sum of logarithms. Suitable for students in Grades 10-12.

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