The given problem is asking to solve for $x$ in the equation:

$\log_{10} x = -2$

Step-by-step Solution:

1. The definition of logarithms states that if $\log_b a = c$, then $a = b^c$.

   • In this case, $b = 10$, $c = -2$, and $a = x$.
   • So, we can rewrite the equation as: $x = 10^{-2}$

2. Calculate $10^{-2}$: $10^{-2} = \frac{1}{100}$

Thus, the value of $x$ is:

$x = \frac{1}{100}$

Conclusion:

The correct answer is the first option: $x = \frac{1}{100}$

Would you like more details on logarithmic functions or have any other questions?

Here are 5 related questions to expand on this topic:

1. What is the logarithmic rule for converting logs to exponents?
2. How do you solve logarithmic equations with different bases?
3. What is the inverse function of a logarithmic equation?
4. How would you handle a logarithmic equation with a variable in the base?
5. What happens if the logarithmic equation is $\log_b x = -1$?

Tip: Logarithms are just another way to express powers! For example, $\log_b x = y$ is the same as saying $x = b^y$.