To expand the logarithmic expression and provide the correct properties, let's go through each step:

Given Expression

$\log_b \frac{nx}{(y - m)^p}$

Step 1

Using the Quotient Rule of logarithms, which states that $\log_b \frac{A}{B} = \log_b A - \log_b B$, we can separate the numerator and the denominator:

$\log_b \frac{nx}{(y - m)^p} = \log_b (nx) - \log_b \left((y - m)^p\right)$

So, the first blank should be filled with Quotient Rule.

Step 2

Now, apply the Product Rule of logarithms, which states that $\log_b (A \cdot B) = \log_b A + \log_b B$, to the term $\log_b (nx)$:

$\log_b (nx) - \log_b \left((y - m)^p\right) = \log_b (n) + \log_b (x) - \log_b \left((y - m)^p\right)$

So, the second blank should be filled with Product Rule.

Step 3

Finally, apply the Power Rule of logarithms, which states that $\log_b (A^c) = c \cdot \log_b A$, to $\log_b \left((y - m)^p\right)$:

$\log_b (n) + \log_b (x) - p \cdot \log_b (y - m)$

So, the third blank should be filled with Power Rule.

Final Answer

• First blank: Quotient Rule
• Second blank: Product Rule
• Third blank: Power Rule

Would you like a deeper explanation of any of these logarithmic properties?

Related Questions

1. How does the Quotient Rule for logarithms differ from the Product Rule?
2. Can logarithmic properties be applied to other bases besides base $b$?
3. What is the purpose of expanding logarithmic expressions in algebra?
4. How does the Power Rule simplify logarithmic expressions with exponents?
5. What are some common mistakes when expanding logarithmic expressions?

Tip:

Always apply logarithmic rules in order: start with Quotient or Product Rules before applying the Power Rule, as this ensures correct simplification.