To solve for [tex]\(\log_{0.5} 16\)[/tex], we need to determine the exponent [tex]\(x\)[/tex] such that [tex]\(0.5^x = 16\)[/tex].
Let's go through the steps to solve for [tex]\(x\)[/tex]:
1. Understand the base and the number: We are given the base [tex]\(0.5\)[/tex] and the number [tex]\(16\)[/tex].
2. Express the logarithmic equation in exponential form: The equation [tex]\(\log_{0.5} 16 = x\)[/tex] can be rewritten as:
[tex]\[
0.5^x = 16
\][/tex]
3. Rewrite 0.5 as a power of 2: Since [tex]\(0.5\)[/tex] is equal to [tex]\(\frac{1}{2}\)[/tex], we can rewrite the equation with base [tex]\(2\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^x = 16
\][/tex]
4. Simplify the base [tex]\(\frac{1}{2}\)[/tex]: The term [tex]\(\frac{1}{2}\)[/tex] can be rewritten as [tex]\(2^{-1}\)[/tex]:
[tex]\[
(2^{-1})^x = 16
\][/tex]
5. Apply the power rule of exponents: When raising a power to another power, multiply the exponents:
[tex]\[
2^{-x} = 16
\][/tex]
6. Rewrite 16 as a power of 2: We know that [tex]\(16 = 2^4\)[/tex], so the equation becomes:
[tex]\[
2^{-x} = 2^4
\][/tex]
7. Set the exponents equal to each other: Since the bases are the same, we can set the exponents equal:
[tex]\[
-x = 4
\][/tex]
8. Solve for [tex]\(x\)[/tex]: We isolate [tex]\(x\)[/tex] by dividing both sides by [tex]\(-1\)[/tex]:
[tex]\[
x = -4
\][/tex]
Thus, the value of [tex]\(\log_{0.5} 16\)[/tex] is [tex]\(-4.0\)[/tex].
Therefore, the correct answer is [tex]\(-4.00\)[/tex].
