Sure, let's go through the step-by-step solution to simplify the given expression:
Given expression:
[tex]\[
\ln \left( \frac{1}{121^k} \right)
\][/tex]
1. Apply the property of logarithms:
The natural logarithm of a quotient can be expressed as the difference of the natural logarithms of the numerator and the denominator:
[tex]\[
\ln \left( \frac{a}{b} \right) = \ln (a) - \ln (b)
\][/tex]
Apply this property to our expression:
[tex]\[
\ln \left( \frac{1}{121^k} \right) = \ln(1) - \ln(121^k)
\][/tex]
2. Simplify [tex]\(\ln(1)\)[/tex]:
We know that [tex]\(\ln(1) = 0\)[/tex]. So, the expression simplifies to:
[tex]\[
0 - \ln(121^k)
\][/tex]
This simplifies to:
[tex]\[
-\ln(121^k)
\][/tex]
3. Apply the power rule of logarithms:
The natural logarithm of an exponentiated quantity can be expressed as the product of the exponent and the natural logarithm of the base:
[tex]\[
\ln(a^b) = b \cdot \ln(a)
\][/tex]
Apply this property to our expression:
[tex]\[
\ln(121^k) = k \cdot \ln(121)
\][/tex]
4. Simplify the original expression:
Substitute [tex]\(\ln(121^k)\)[/tex] with [tex]\(k \cdot \ln(121)\)[/tex] in the original expression:
[tex]\[
-\ln(121^k) = - \left( k \cdot \ln(121) \right)
\][/tex]
This results in:
[tex]\[
-k \cdot \ln(121)
\][/tex]
So, the simplified expression is:
[tex]\[
\ln \left( \frac{1}{121^k} \right) = -k \cdot \ln(121)
\][/tex]
