To simplify the given logarithmic expression step by step, we can use the properties of logarithms. The expression is:

$\frac{1}{2} \log_b x + 2 \log_b 6 - 10 \log_b y - 5 \log_b z$

We will use the following properties of logarithms:

1. $a \log_b c = \log_b (c^a)$
2. $\log_b m + \log_b n = \log_b (mn)$
3. $\log_b m - \log_b n = \log_b \left( \frac{m}{n} \right)$

Step-by-Step Solution:

1. Apply the power rule to each term:

$\frac{1}{2} \log_b x = \log_b x^{\frac{1}{2}}$

$2 \log_b 6 = \log_b 6^2 = \log_b 36$

$10 \log_b y = \log_b y^{10}$

$5 \log_b z = \log_b z^5$

2. Rewrite the expression using these transformations:

$\log_b x^{\frac{1}{2}} + \log_b 36 - \log_b y^{10} - \log_b z^5$

3. Combine the logarithms using the product and quotient rules:

First, combine the addition terms:

$\log_b x^{\frac{1}{2}} + \log_b 36 = \log_b \left( x^{\frac{1}{2}} \cdot 36 \right)$

Next, combine the subtraction terms:

$\log_b \left( x^{\frac{1}{2}} \cdot 36 \right) - \log_b y^{10} = \log_b \left( \frac{x^{\frac{1}{2}} \cdot 36}{y^{10}} \right)$

Then:

$\log_b \left( \frac{x^{\frac{1}{2}} \cdot 36}{y^{10}} \right) - \log_b z^5 = \log_b \left( \frac{x^{\frac{1}{2}} \cdot 36}{y^{10} \cdot z^5} \right)$

4. Simplify the expression:

$\log_b \left( \frac{36 \sqrt{x}}{y^{10} z^5} \right)$

So, the simplified form of the given expression is:

$\log_b \left( \frac{36 \sqrt{x}}{y^{10} z^5} \right)$

Would you like any further details or have any questions about this solution?

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Follow-Up Questions:

1. What is the power rule for logarithms and how is it applied?
2. How can the product rule for logarithms be used to combine multiple logarithmic terms?
3. What is the quotient rule for logarithms and when is it useful?
4. How can we simplify the expression $\log_b x^{\frac{1}{2}}$?
5. Can you explain the step-by-step process of combining logarithms?
6. What is the importance of simplifying logarithmic expressions in mathematics?
7. How would you handle the expression if there were additional terms with different bases?
8. Can you convert the final logarithmic expression back to an exponential form?

Tip:

When simplifying logarithmic expressions, always look for opportunities to apply the logarithmic properties systematically, which will help in combining and simplifying the terms effectively.