Solving Logarithmic Equations with Multiple Steps
To solve the equation involving logarithms from the image provided, follow these steps:
Given the equation:
\[ \frac{\log_2{3a} + \log_2{16}}{\log_2{x}} = \log_2{x} \]
1. First apply the log rule \(\log_b{m} + \log_b{n} = \log_b{mn}\) to combine the logs in the numerator:
\[ \frac{\log_2{(3a \cdot 16)}}{\log_2{x}} = \log_2{x} \]
2. Since \(16 = 2^4\), we can write the equation as:
\[ \frac{\log_2{(3a \cdot 2^4)}}{\log_2{x}} = \log_2{x} \]
\[ \frac{\log_2{48a}}{\log_2{x}} = \log_2{x} \]
3. Multiply both sides by \(\log_2{x}\) to get rid of the denominator:
\[ \log_2{48a} = (\log_2{x})^2 \]
4. Recognize that \((\log_2{x})^2\) is the same as \(\log_2{x^2}\), and use the property that \(\log_b{m} = \log_b{n}\) implies \(m = n\):
\[ 48a = x^2 \]
5. You can now solve for \(a\), given \(x\), or solve for \(x\) given \(a\). If you want to find a formula for \(a\) in terms of \(x\), divide by 48:
\[ a = \frac{x^2}{48} \]
Or to find \(x\) in terms of \(a\), take the square root of both sides:
\[ x = \pm\sqrt{48a} \]
In the context of logarithms, \(x\) is typically assumed to be positive, so the negative solution is often discarded, leaving:
\[ x = \sqrt{48a} \]
