Answer :
Certainly! Let's condense the given logarithmic expression using the properties of logarithms step by step.
Given expression:
[tex]\[ 6 \ln(x) - \frac{1}{4} \ln(y) \][/tex]
### Step 1: Apply the power rule of logarithms
The power rule states that \( a \ln(b) \) can be rewritten as \( \ln(b^a) \). We will apply this rule to each term in the expression.
#### For the first term \( 6 \ln(x) \):
[tex]\[ 6 \ln(x) = \ln(x^6) \][/tex]
#### For the second term \( -\frac{1}{4} \ln(y) \):
[tex]\[ -\frac{1}{4} \ln(y) = \ln(y^{-\frac{1}{4}}) \][/tex]
So now the expression looks like:
[tex]\[ \ln(x^6) - \ln(y^{\frac{1}{4}}) \][/tex]
### Step 2: Apply the quotient rule of logarithms
The quotient rule states that \( \ln(a) - \ln(b) \) can be rewritten as \( \ln\left(\frac{a}{b}\right) \). We will apply this rule to the current expression we have.
[tex]\[ \ln(x^6) - \ln(y^{\frac{1}{4}}) = \ln\left(\frac{x^6}{y^{\frac{1}{4}}}\right) \][/tex]
Thus, the condensed logarithmic expression is:
[tex]\[ \ln\left(\frac{x^6}{y^{\frac{1}{4}}}\right) \][/tex]
Therefore, the final condensed form of the given expression \( 6 \ln(x) - \frac{1}{4} \ln(y) \) is:
[tex]\[ \boxed{\ln\left(\frac{x^6}{y^{\frac{1}{4}}}\right)} \][/tex]
Given expression:
[tex]\[ 6 \ln(x) - \frac{1}{4} \ln(y) \][/tex]
### Step 1: Apply the power rule of logarithms
The power rule states that \( a \ln(b) \) can be rewritten as \( \ln(b^a) \). We will apply this rule to each term in the expression.
#### For the first term \( 6 \ln(x) \):
[tex]\[ 6 \ln(x) = \ln(x^6) \][/tex]
#### For the second term \( -\frac{1}{4} \ln(y) \):
[tex]\[ -\frac{1}{4} \ln(y) = \ln(y^{-\frac{1}{4}}) \][/tex]
So now the expression looks like:
[tex]\[ \ln(x^6) - \ln(y^{\frac{1}{4}}) \][/tex]
### Step 2: Apply the quotient rule of logarithms
The quotient rule states that \( \ln(a) - \ln(b) \) can be rewritten as \( \ln\left(\frac{a}{b}\right) \). We will apply this rule to the current expression we have.
[tex]\[ \ln(x^6) - \ln(y^{\frac{1}{4}}) = \ln\left(\frac{x^6}{y^{\frac{1}{4}}}\right) \][/tex]
Thus, the condensed logarithmic expression is:
[tex]\[ \ln\left(\frac{x^6}{y^{\frac{1}{4}}}\right) \][/tex]
Therefore, the final condensed form of the given expression \( 6 \ln(x) - \frac{1}{4} \ln(y) \) is:
[tex]\[ \boxed{\ln\left(\frac{x^6}{y^{\frac{1}{4}}}\right)} \][/tex]
