Question

What is the domain of f(x) = log(5 − 2x)?

Answer 1

The domain of
`\( f(x) = \log(5 - 2x) \)` is
`\( x \leq 2.5 \)` or
`\( (-\infty, 2.5] \).`

Step-by-step explanation:

The logarithm function
`\( \log \)` is defined only for positive arguments. In this case, the argument inside the logarithm,
`\( (5 - 2x) \)`, must be greater than zero. So, we set the inequality
`\( 5 - 2x > 0 \)` and solve for ( x):

`\[ 5 - 2x > 0 \]`

Subtract 5 from both sides:

`\[ -2x > -5 \]`

Divide both sides by -2 (note that dividing by a negative number reverses the inequality sign):

`\[ x < (5)/(2) \]`

So, the domain is
`\( x < (5)/(2) \)`or, in interval notation,
`\( (-\infty, 2.5] \)`.

It's important to note that the domain is inclusive of
`\( x = 2.5 \)` because at
`\( x = 2.5 \)`, the argument
`\( (5 - 2 * 2.5) \)` becomes zero, making the logarithm defined. Beyond ( x = 2.5), the argument becomes negative, leading to an undefined logarithm, and thus,
`\( x \leq 2.5 \).`