Let's solve the problem shown in the image step by step:

The problem is:
$\log{\frac{1}{1000}}$

Step 1: Simplify the fraction

The expression $\frac{1}{1000}$ can be rewritten as $1000^{-1}$. So,
$\log{\frac{1}{1000}} = \log{(1000^{-1})}.$

Step 2: Use the logarithmic power rule

The logarithmic power rule states:
$\log{(a^b)} = b \cdot \log{a}.$
Applying this,
$\log{(1000^{-1})} = -1 \cdot \log{1000}.$

Step 3: Compute $\log{1000}$

Assuming the logarithm is base 10 ($\log_{10}$), we know that $1000 = 10^3$. Thus:
$\log{1000} = 3.$

Step 4: Combine results

Substitute $\log{1000} = 3$ into the earlier result:
$-1 \cdot \log{1000} = -1 \cdot 3 = -3.$

Final Answer:

$\log{\frac{1}{1000}} = -3.$

Let me know if you’d like further details or have questions!

Related Questions:

1. How would you compute $\log_{2}{\frac{1}{16}}$?
2. What is $\log{100}$ using base 10?
3. How do logarithm rules apply when adding two logarithms?
4. What is the value of $\ln{\frac{1}{e^2}}$?
5. How would the solution change if the base of the logarithm were different?

Tip:

When simplifying logarithmic expressions, remember that negative exponents translate into division (e.g., $10^{-1} = \frac{1}{10}$). This makes it easier to handle fractions!