\ln\left(\sqrt{15x^{2}+11}\right)-\ln\left(x\right)

The expression you've provided is:

\[
\ln\left(\sqrt{15x^{2} + 11}\right) - \ln\left(x\right)
\]

We can simplify this using logarithmic properties. First, recall the following rules of logarithms:

1. \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\)
2. \(\ln\left(\sqrt{A}\right) = \frac{1}{2}\ln(A)\)

### Step 1: Apply the logarithm of a square root
\[
\ln\left(\sqrt{15x^{2} + 11}\right) = \frac{1}{2}\ln(15x^{2} + 11)
\]

### Step 2: Combine the two logarithms
Now, using the difference of logarithms rule:

\[
\frac{1}{2} \ln(15x^{2} + 11) - \ln(x) = \ln\left(\frac{\sqrt{15x^{2} + 11}}{x}\right)
\]

So, the simplified form is:

\[
\ln\left(\frac{\sqrt{15x^{2} + 11}}{x}\right)
\]

Would you like more details or clarification on any steps?

Here are 5 related questions to deepen your understanding:

1. How would this expression change if we had \(\ln\left(15x^{2} + 11\right)\) instead of the square root?
2. Can you solve for \(x\) if \(\ln\left(\frac{\sqrt{15x^{2} + 11}}{x}\right) = 0\)?
3. What happens to the logarithmic expression as \(x\) approaches 0?
4. How would you differentiate the expression \(\ln\left(\frac{\sqrt{15x^{2} + 11}}{x}\right)\)?
5. What is the domain of the function \(\ln\left(\sqrt{15x^{2} + 11}\right) - \ln\left(x\right)\)?

**Tip:** Remember that \(\ln(x)\) is only defined for positive values of \(x\).

This solution demonstrates how to simplify the logarithmic expression ln(√(15x^2 + 11)) - ln(x) using key logarithmic properties. Follow step-by-step instructions on how to apply logarithmic rules and simplifications, suitable for high school students in Grades 10-12.

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