Are the domains of logarithmic functions all real numbers?

Logarithmic functions do not have all real numbers as their domain. Instead, the domain of a logarithmic function is limited to positive real numbers.

To understand this better, consider the basic form of a logarithmic function, which is f(x) = log_b(x), where b is the base of the logarithm and x is the argument. For the logarithmic function to return a real number, the argument x must be greater than zero (x > 0). This is because the logarithm of a non-positive number (zero or negative) is undefined in the realm of real numbers.

For example:

  • log_2(1) is defined and equals 0.
  • log_2(2) is defined and equals 1.
  • log_2(0) is undefined.
  • log_2(-1) is also undefined.

Thus, the domain of any logarithmic function consists exclusively of all positive real numbers, represented as (0, +∞). So, to conclude, no, the domains of logarithmic functions are not all real numbers; they are specifically confined to positive real numbers.

More Related Questions