To find the value of the logarithm [tex]\(\log_2 \left( \frac{1}{50} \right)\)[/tex], we can approach this problem with the following steps:
1. Express the value using a base of 2:
The expression [tex]\(\frac{1}{50}\)[/tex] can be written directly as is because it represents [tex]\(\frac{1}{50}\)[/tex].
2. Understanding the logarithm:
The expression [tex]\(\log_2 \left( \frac{1}{50} \right)\)[/tex] asks us for the power to which the base [tex]\(2\)[/tex] must be raised to obtain [tex]\(\frac{1}{50}\)[/tex].
3. Approximate the value:
Using logarithmic properties and tools, we find that:
[tex]\[
\log_2 \left( \frac{1}{50} \right) \approx -5.643856189774724
\][/tex]
4. Round to the nearest thousandth:
To provide the answer rounded to the nearest thousandth, we take the value -5.643856189774724 and round it appropriately. The thousandth place is three decimal places to the right of the decimal point.
[tex]\[
\log_2 \left( \frac{1}{50} \right) \approx -5.644
\][/tex]
So, the value of [tex]\(\log_2 \left( \frac{1}{50} \right)\)[/tex] rounded to the nearest thousandth is approximately [tex]\( -5.644 \)[/tex].
