Question

Is the range of a logarithmic function always all real numbers?.

Answer 1

Answer: Yes

Step-by-step explanation:

The log equation
`\text{x} = \log_b(\text{y})` is equivalent to its exponential form
`\text{y} = b^\text{x}`

For the exponential equation, we can replace x with any real number. We don't have to worry about things like dividing by zero. Therefore the domain of the exponential equation is "set of all real numbers".

The domain of
`\text{y} = b^\text{x}` is the range of
`\text{x} = \log_b(\text{y})`. The domain and range swap roles when going from original to inverse.

Therefore, the range of any logarithmic function is "set of all real numbers".

On a graph notice how the log curve stretches forever upward and downward. The base of the log doesn't matter.