At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.