We are solving the equation:

$\log{x} = -\log{2}.$

Step 1: Simplify the equation

Using the property of logarithms $-\log{a} = \log{\frac{1}{a}}$, rewrite the equation as:

$\log{x} = \log{\frac{1}{2}}.$

Step 2: Eliminate the logarithm

Since the logarithms on both sides have the same base (assumed to be 10 if not specified), their arguments must be equal:

$x = \frac{1}{2}.$

Final Solution:

$x = \frac{1}{2}.$

Explanation:

• The negative sign in front of $\log{2}$ flips the argument to its reciprocal ($\frac{1}{2}$).
• Since logarithms are equal when their arguments are equal, we directly equate $x$ to $\frac{1}{2}$.

Would you like further clarification or more examples? Here are 5 related questions to explore:

1. What is the general property of logarithms when the argument is negative or a fraction?
2. How do logarithmic properties simplify solving equations with multiple terms?
3. Can you use natural logarithms ($\ln$) to solve the same problem?
4. What is the inverse operation of a logarithm, and how does it help in equations?
5. How does the base of the logarithm affect solving such equations?

Tip: Always check if the argument of a logarithm is positive, as $\log(x)$ is only defined for $x > 0$.