To combine the expression into a single logarithm with a coefficient of 1, let's go through each term and use the properties of logarithms.

Given Expression

$\ln(2x + 7) - \ln y - 6 \ln z + \frac{1}{2} \ln w$

Step 1: Apply the Power Rule of Logarithms

The power rule of logarithms states that $a \ln b = \ln(b^a)$. We apply this rule to the terms $-6 \ln z$ and $\frac{1}{2} \ln w$.

$\ln(2x + 7) - \ln y - \ln(z^6) + \ln(w^{1/2})$

Step 2: Rewrite $\ln(w^{1/2})$ as $\ln(\sqrt{w})$

Using the fact that $w^{1/2} = \sqrt{w}$, we can rewrite the term as:

$\ln(2x + 7) - \ln y - \ln(z^6) + \ln(\sqrt{w})$

Step 3: Combine Terms Using the Quotient and Product Rules

The quotient rule of logarithms states that $\ln a - \ln b = \ln\left(\frac{a}{b}\right)$. The product rule states that $\ln a + \ln b = \ln(ab)$.

So we combine the terms as follows:

$\ln \left(\frac{(2x + 7) \sqrt{w}}{y z^6}\right)$

Final Answer

The expression as a single logarithm is:

$\ln \left(\frac{(2x + 7) \sqrt{w}}{y z^6}\right)$

Would you like further details on the properties used here, or do you have any other questions?

Here are some related questions:

1. What is the power rule of logarithms, and how is it used?
2. How do the product and quotient rules of logarithms work?
3. Can logarithmic expressions be simplified when variables include exponents?
4. How does the assumption of positive real numbers impact the simplification?
5. What happens if negative values are introduced in logarithmic terms?

Tip: Always verify that all terms are positive when combining logarithmic expressions, as logarithms of non-positive numbers are undefined in the real number system.