To find the domain of the function $\log(x)$, we need to determine the set of all values of $x$ for which the expression is valid.

Step-by-Step Process:

The logarithmic function $\log(x)$ is only defined for positive real numbers. In other words, the argument of the logarithm must be strictly greater than zero.

Thus, the function $\log(x)$ is defined when: $x > 0$

Domain:

Therefore, the domain of $\log(x)$ is all positive real numbers, which can be expressed as: $\boxed{(0, \infty)}$

Additional Clarification:

• For $x = 0$, $\log(0)$ is undefined because there is no number $y$ such that $10^y = 0$.
• For $x < 0$, $\log(x)$ is also undefined in the realm of real numbers, because logarithms of negative numbers don't exist in the real number system.

Would you like further details on the properties of logarithmic functions, or any related concepts?

Related Questions:

1. What is the domain of $\log_b(x)$ for any base $b > 0$?
2. How does the domain of logarithmic functions change when the base is different from 10 (like $\ln(x)$)?
3. What happens to the logarithmic function as $x$ approaches zero from the positive side?
4. How can logarithmic functions be transformed (e.g., shifting or scaling)?
5. How do you graph $\log(x)$?

Tip:

When working with logarithms, always remember that their domain is restricted to positive real numbers. If you need to evaluate a logarithmic expression, first check that the argument is positive!