Question

How do you find the domain and range of `y = log(2x -12)`?

Answer 1

For domain, the argument of the logarithm must be greater than 0.
Domain: `\{x | x > 6, x \in \mathbb{R}}`
Range: `\{y | y \in \mathbb{R}}`

Explanation:

For domain:

The argument of the logarithm (stuff inside the log) must be greater than 0. This is because logarithm can be viewed as the inverse of an exponential function. For example:

`y=\log(2x-12)`
`y = \log_{10}(2x-12)`

This can be represented by, in exponential form,

`10^y = 2x - 12`

10 raised to any exponent cannot get a negative number or be equal to zero, thus `2x - 12 > 0`

So solve `2x - 12 > 0` to get the permissible values for `x`.

`2x - 12 > 0`
`\ \ \ \ \ \ \ \ \ 2x > 12`
`\ \ \ \ \ \ \ \ \ \ \ x > 6`

The domain is `\{x | x > 6, x \in \mathbb{R}}`

For range

For any logarithmic function of the form `y = a \log_c(b(x-h))+k`, the range is `\{y | y \in \mathbb{R}}`.