Answer :
To condense the logarithmic expression \( 6 \log_b q - \log_b r \), we can use the properties of logarithms. Let’s go through the steps in detail.
### Step 1: Apply the Power Rule
First, we use the power rule of logarithms, which states that \( a \log_b x = \log_b (x^a) \).
Applying this to the term \( 6 \log_b q \):
[tex]\[ 6 \log_b q = \log_b (q^6) \][/tex]
### Step 2: Combine the Logarithms Using the Quotient Rule
Next, we use the quotient rule of logarithms, which states that \( \log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) \).
Substitute \( \log_b (q^6) \) for \( 6 \log_b q \) and combine it with the term \( -\log_b r \):
[tex]\[ \log_b (q^6) - \log_b r = \log_b \left( \frac{q^6}{r} \right) \][/tex]
### Final Expression
Therefore, the condensed form of the expression \( 6 \log_b q - \log_b r \) is:
[tex]\[ \log_b \left( \frac{q^6}{r} \right) \][/tex]
So, the final condensed expression is:
[tex]\[ \boxed{\log_b \left( \frac{q^6}{r} \right)} \][/tex]
### Step 1: Apply the Power Rule
First, we use the power rule of logarithms, which states that \( a \log_b x = \log_b (x^a) \).
Applying this to the term \( 6 \log_b q \):
[tex]\[ 6 \log_b q = \log_b (q^6) \][/tex]
### Step 2: Combine the Logarithms Using the Quotient Rule
Next, we use the quotient rule of logarithms, which states that \( \log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) \).
Substitute \( \log_b (q^6) \) for \( 6 \log_b q \) and combine it with the term \( -\log_b r \):
[tex]\[ \log_b (q^6) - \log_b r = \log_b \left( \frac{q^6}{r} \right) \][/tex]
### Final Expression
Therefore, the condensed form of the expression \( 6 \log_b q - \log_b r \) is:
[tex]\[ \log_b \left( \frac{q^6}{r} \right) \][/tex]
So, the final condensed expression is:
[tex]\[ \boxed{\log_b \left( \frac{q^6}{r} \right)} \][/tex]
