Question - Solving Logarithmic Equations with Multiple Steps
Solution:
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:\[ x = \sqrt{48a} \]
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