Solving Logarithmic Expressions with Properties
To solve the expression given in the image as a single logarithm, we can use the properties of logarithms:
The original expression given is:
\[ \frac{1}{4} \ln 4 - 2 \ln x \]
Firstly, we can use the power rule of logarithms which states that \( \log_b (a^n) = n \log_b a \). We can apply this rule on the first term to bring down the 1/4 as a power of 4 inside the logarithm:
\[ = \ln 4^{1/4} - 2 \ln x \]
\[ = \ln 2 - 2 \ln x \]
(Note: \( 4^{1/4} \) is the fourth root of 4, which is equal to 2.)
Next, we use another property of logarithms: the coefficient in front of a logarithm can be converted into a power inside the logarithm, which is:
\[ \log_b a^c = c \log_b a \]
Applying this to the second term:
\[ = \ln 2 - \ln x^2 \]
Now, we use the subtraction rule for logarithms, which states: \( \log_b a - \log_b c = \log_b \frac{a}{c} \). We'll apply this to the expression:
\[ = \ln \frac{2}{x^2} \]
Looking at the options provided in the image, the one that matches is B:
\[ \ln \left( \frac{2}{x^2} \right) \]
