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 equals0.log_2(2)is defined and equals1.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.