https://file.aiquickdraw.com/mathbot/file/1726530430508pinu0iw2.jpg

Let's break down and solve the problems shown in the image step by step.

### (a) Expanding \( \log_{10}(x^5 y^4 z) \)

We can use the properties of logarithms, especially the **product rule** and the **power rule**:

- **Product rule**: \( \log_b(abc) = \log_b(a) + \log_b(b) + \log_b(c) \)
- **Power rule**: \( \log_b(a^n) = n \log_b(a) \)

Given the expression \( \log_{10}(x^5 y^4 z) \), we apply the product rule and then the power rule to each term:

\[
\log_{10}(x^5 y^4 z) = \log_{10}(x^5) + \log_{10}(y^4) + \log_{10}(z)
\]

Now, applying the power rule to the first two terms:

\[
= 5 \log_{10}(x) + 4 \log_{10}(y) + \log_{10}(z)
\]

Thus, the expanded form is:

\[
5 \log_{10}(x) + 4 \log_{10}(y) + \log_{10}(z)
\]

This matches the provided solution in the image.

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### (b) Expanding \( \ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right), \text{ where } x > 6 \)

Here, we need to use both the **quotient rule** and the **power rule** for logarithms:

- **Quotient rule**: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
- **Power rule**: \( \ln(a^n) = n \ln(a) \)

Now, applying these to the given expression:

1. Use the quotient rule:

\[
\ln \left( \frac{x^3}{\sqrt{x^2 - 36}} \right) = \ln(x^3) - \ln(\sqrt{x^2 - 36})
\]

2. Apply the power rule to both terms:

\[
= 3 \ln(x) - \frac{1}{2} \ln(x^2 - 36)
\]

So, the fully expanded form is:

\[
3 \ln(x) - \frac{1}{2} \ln(x^2 - 36)
\]

That is the answer for part (b).

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Would you like more details or have any other questions?

### 5 Related Questions:
1. How would you condense logarithmic expressions using the laws of logarithms?
2. What are the key differences between common logarithms and natural logarithms?
3. Can you explain the change of base formula for logarithms?
4. How would you differentiate logarithmic functions like \( \log_b(x) \) or \( \ln(x) \)?
5. What are the typical real-world applications of logarithmic functions in science and engineering?

### Tip:
Always remember that logarithmic rules simplify complex multiplication, division, or exponentiation, making calculations more manageable.

Use the laws of logarithms to expand the expression: (a) log10(x^5 * y^4 * z), (b) ln(x^3 / √(x^2 - 36)), x > 6.

This problem explores logarithmic expansion using the product, quotient, and power rules. Learn how to expand log10(x^5 * y^4 * z) and ln(x^3 / √(x^2 - 36)), applying key logarithmic principles.

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