Solving a Logarithmic Expression Step by Step
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 \(\log_2(3)\).
