Question

for the exact value of x.
\\(\log_{2}(2x) + 2\log_{2}(4) = 4\\)

Explanation:

Step1: Simplify \(2\log_{2}(4)\)

We know that \(\log_{a}(a^{b}) = b\). Since \(4 = 2^{2}\), then \(\log_{2}(4)=\log_{2}(2^{2}) = 2\). So \(2\log_{2}(4)=2\times2 = 4\).
The equation becomes \(\log_{2}(2x)+4 = 4\).

Step2: Isolate the logarithmic term

Subtract 4 from both sides of the equation: \(\log_{2}(2x)+4 - 4=4 - 4\), which simplifies to \(\log_{2}(2x)=0\).

Step3: Convert the logarithmic equation to exponential form

Recall that if \(\log_{a}(b)=c\), then \(a^{c}=b\). Here, \(a = 2\), \(c = 0\), and \(b = 2x\). So \(2^{0}=2x\).
Since \(2^{0}=1\), we have \(1 = 2x\).

Step4: Solve for \(x\)

Divide both sides of the equation \(1 = 2x\) by 2: \(x=\frac{1}{2}\).

Answer:

\(x = \frac{1}{2}\)