To solve for [tex]\( x \)[/tex] in the equation [tex]\( x = \log_2 \sqrt{2} \)[/tex], we need to manipulate the expression inside the logarithm to more easily determine its value.
Step-by-Step Solution:
1. Recognize the Square Root:
[tex]\[
\sqrt{2}
\][/tex]
The square root of 2 can be expressed as a power of 2.
[tex]\[
\sqrt{2} = 2^{1/2}
\][/tex]
2. Substitute the Expression:
[tex]\[
x = \log_2 (\sqrt{2}) = \log_2 (2^{1/2})
\][/tex]
3. Use the Logarithm Power Rule:
One of the properties of logarithms states that [tex]\( \log_b (a^c) = c \cdot \log_b a \)[/tex].
Applying this property:
[tex]\[
\log_2 (2^{1/2}) = \frac{1}{2} \log_2 (2)
\][/tex]
4. Simplify the Logarithm:
The logarithm of a number to its own base is always 1:
[tex]\[
\log_2 (2) = 1
\][/tex]
5. Calculate the Value:
Multiply by the coefficient outside of the logarithm:
[tex]\[
x = \frac{1}{2} \cdot 1 = \frac{1}{2}
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Answer:
\boxed{\frac{1}{2}}
