Answer :
Certainly! Let's rewrite the given expression [tex]\(2 \log_4(x) - 7 \log_4(y)\)[/tex] using properties of logarithms.
### Step-by-Step Solution:
1. Identify the given expression:
[tex]\[ 2 \log_4(x) - 7 \log_4(y) \][/tex]
2. Apply the power rule of logarithms:
The power rule states that [tex]\(a \log_b(c) = \log_b(c^a)\)[/tex]. Using this property, we can transform each term individually:
- For the first term [tex]\(2 \log_4(x)\)[/tex]:
[tex]\[ 2 \log_4(x) = \log_4(x^2) \][/tex]
- For the second term [tex]\(7 \log_4(y)\)[/tex]:
[tex]\[ 7 \log_4(y) = \log_4(y^7) \][/tex]
3. Rewrite the expression using the results from the power rule:
After applying the power rule, the expression becomes:
[tex]\[ \log_4(x^2) - \log_4(y^7) \][/tex]
4. Apply the subtraction property of logarithms:
The subtraction property, also known as the quotient rule, states that [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex]. Using this property, we can combine the two logarithms into a single logarithm:
[tex]\[ \log_4(x^2) - \log_4(y^7) = \log_4\left(\frac{x^2}{y^7}\right) \][/tex]
5. Write the final simplified expression:
Therefore, the given expression [tex]\(2 \log_4(x) - 7 \log_4(y)\)[/tex] can be rewritten as:
[tex]\[ \log_4\left(\frac{x^2}{y^7}\right) \][/tex]
So, the simplified form is:
[tex]\[ 2 \log_4(x) - 7 \log_4(y) = \log_4\left(\frac{x^2}{y^7}\right) \][/tex]
### Step-by-Step Solution:
1. Identify the given expression:
[tex]\[ 2 \log_4(x) - 7 \log_4(y) \][/tex]
2. Apply the power rule of logarithms:
The power rule states that [tex]\(a \log_b(c) = \log_b(c^a)\)[/tex]. Using this property, we can transform each term individually:
- For the first term [tex]\(2 \log_4(x)\)[/tex]:
[tex]\[ 2 \log_4(x) = \log_4(x^2) \][/tex]
- For the second term [tex]\(7 \log_4(y)\)[/tex]:
[tex]\[ 7 \log_4(y) = \log_4(y^7) \][/tex]
3. Rewrite the expression using the results from the power rule:
After applying the power rule, the expression becomes:
[tex]\[ \log_4(x^2) - \log_4(y^7) \][/tex]
4. Apply the subtraction property of logarithms:
The subtraction property, also known as the quotient rule, states that [tex]\(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\)[/tex]. Using this property, we can combine the two logarithms into a single logarithm:
[tex]\[ \log_4(x^2) - \log_4(y^7) = \log_4\left(\frac{x^2}{y^7}\right) \][/tex]
5. Write the final simplified expression:
Therefore, the given expression [tex]\(2 \log_4(x) - 7 \log_4(y)\)[/tex] can be rewritten as:
[tex]\[ \log_4\left(\frac{x^2}{y^7}\right) \][/tex]
So, the simplified form is:
[tex]\[ 2 \log_4(x) - 7 \log_4(y) = \log_4\left(\frac{x^2}{y^7}\right) \][/tex]
