Question - Solving Logarithmic Expressions with Properties
mathlogarithmic propertiessolving logarithmic expressionspower rule of logarithmssubtraction rule for logarithms
Solution:
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:\[ \ln \left( \frac{2}{x^2} \right) \]
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