Certainly! Let's start by condensing the given logarithmic expression step by step.
Given expression:
[tex]\[ 3 \left( \ln \left( \sqrt[3]{e^7} \right) - \ln (xy) \right) \][/tex]
### Step 1: Simplify the term inside the logarithm
[tex]\[ \sqrt[3]{e^7} \][/tex]
This is the cube root of [tex]\(e^7\)[/tex], which can be rewritten using exponents:
[tex]\[ \sqrt[3]{e^7} = (e^7)^{1/3} = e^{7/3} \][/tex]
Now substitute this back into the logarithm:
[tex]\[ \ln \left( \sqrt[3]{e^7} \right) = \ln \left( e^{7/3} \right) \][/tex]
### Step 2: Apply the properties of logarithms
Using the property of logarithms that [tex]\(\ln (a^b) = b \ln (a)\)[/tex]:
[tex]\[ \ln \left( e^{7/3} \right) = \frac{7}{3} \ln (e) \][/tex]
Since [tex]\(\ln (e) = 1\)[/tex]:
[tex]\[ \ln \left( e^{7/3} \right) = \frac{7}{3} \][/tex]
### Step 3: Substitute back and use the properties of logarithms
Now we have:
[tex]\[ 3 \left( \frac{7}{3} - \ln (xy) \right) \][/tex]
Apply the logarithm property [tex]\(\ln(a \cdot b) = \ln(a) + \ln(b)\)[/tex]:
[tex]\[ \ln (xy) = \ln (x) + \ln (y) \][/tex]
So,
[tex]\[ 3 \left( \frac{7}{3} - [\ln (x) + \ln (y)] \right) \][/tex]
### Step 4: Distribute the 3
[tex]\[ 3 \left( \frac{7}{3} - \ln (x) - \ln (y) \right) \][/tex]
[tex]\[ 3 \cdot \frac{7}{3} - 3 \cdot \ln (x) - 3 \cdot \ln (y) \][/tex]
This simplifies to:
[tex]\[ 7 - 3 \ln (x) - 3 \ln (y) \][/tex]
### Step 5: Combine the logarithms with coefficients
Using the property [tex]\(a \ln(b) = \ln(b^a)\)[/tex], we get:
[tex]\[ -3 \ln (x) = \ln (x^{-3}) \][/tex]
[tex]\[ -3 \ln (y) = \ln (y^{-3}) \][/tex]
Thus, we can write:
[tex]\[ 7 - \ln (x^3) - \ln (y^3) \][/tex]
### Step 6: Combine the logarithms
Using the property [tex]\(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\)[/tex]:
[tex]\[ \ln (x^3) + \ln (y^3) = \ln (x^3 y^3) = \ln \left( (xy)^3 \right) \][/tex]
Finally, we have:
[tex]\[ 7 - \ln \left( (xy)^3 \right) \][/tex]
### Answer
[tex]\[ 7 - \ln \left( (xy)^3 \right) \][/tex]
This is the most condensed form of the given logarithmic expression with all exponents positive and written as a single term with a coefficient of 1.
