Question - Solving a Logarithmic Expression Step by Step
Solution:
The image shows a mathematical expression involving logarithms. We will solve the expression step by step:The expression is:\[ \frac{1}{2} \log_2 \left( \frac{4}{3} \right) - 2\log_2 \left( \frac{9}{4} \right) \]First, let's simplify the logarithmic terms using the properties of logarithms:1) Use the power rule of logarithms, which states that $$\log_b(m^n) = n\cdot\log_b(m)$$, to simplify the second term:\[ 2\log_2 \left( \frac{9}{4} \right) = \log_2 \left( \left(\frac{9}{4}\right)^2 \right) \]2) Calculate the square of $$\frac{9}{4}$$:\[ \left(\frac{9}{4}\right)^2 = \frac{81}{16} \]Now, we can rewrite the expression as:\[ \frac{1}{2} \log_2 \left( \frac{4}{3} \right) - \log_2 \left( \frac{81}{16} \right) \]3) Apply the quotient rule for logarithms, which states that $$\log_b \left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$$, to both logarithmic terms:\[ \frac{1}{2} \left( \log_2(4) - \log_2(3) \right) - \left( \log_2(81) - \log_2(16) \right) \]4) Simplify the logarithms using the fact that $$\log_2(4) = 2$$, $$\log_2(16) = 4$$, and $$\log_2(81)$$ is $$\log_2(3^4)$$ which equals $$4\log_2(3)$$:\[ \frac{1}{2} \left( 2 - \log_2(3) \right) - \left( 4\log_2(3) - 4 \right) \]5) Simplify the expression by distributing and combining like terms:\[ 1 - \frac{1}{2}\log_2(3) - 4\log_2(3) + 4 \]\[ 5 - \left( \frac{1}{2}\log_2(3) + 4\log_2(3) \right) \]\[ 5 - \left( \frac{1}{2} + 4\right) \log_2(3) \]\[ 5 - \frac{9}{2} \log_2(3) \]Here is the simplified form of the original expression:\[ 5 - \frac{9}{2} \log_2(3) \]This is as simplified as the expression can get without knowing the numerical value of $$\log_2(3)$$.
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