Math  /  Algebra

QuestionFind the domain of the function. f(x)=log(5x8)f(x)=\log \left(\frac{-5}{x-8}\right)
Write your answer as an interval or union of intervals.
Domain: \square

Studdy Solution

STEP 1

What is this asking? We need to find all the allowed xx values for the function f(x)=log(5x8)f(x) = \log\left(\frac{-5}{x-8}\right). Watch out! Remember that we can only take the logarithm of **positive numbers**!

STEP 2

1. Analyze the argument of the logarithm.
2. Determine the allowed values for xx.

STEP 3

Alright, so we've got this function f(x)=log(5x8)f(x) = \log\left(\frac{-5}{x-8}\right), and we want to find its **domain**.
The domain of a function is just all the **valid inputs**, or all the xx values we're allowed to plug in.

STEP 4

Now, with logarithms, the **argument** (the stuff inside the parentheses) *must* be **positive**, greater than zero!
So, we need 5x8>0\frac{-5}{x-8} > 0.

STEP 5

Let's think about this.
We have a **negative number** in the numerator (5-5).
For the whole fraction to be positive, the denominator (x8)(x-8) *must* also be **negative**.
A negative divided by a negative gives us a positive!

STEP 6

So, we need x8<0x-8 < 0.
Let's **isolate** xx by adding 88 to both sides of the inequality.

STEP 7

x8+8<0+8x - 8 + 8 < 0 + 8 x<8x < 8

STEP 8

Also, remember that we can't divide by zero, so x8x-8 cannot be zero.
This means xx cannot be 88, which we already took care of with our inequality x<8x < 8.

STEP 9

So, xx can be any number **less than** 88.

STEP 10

Domain: (,8)(-\infty, 8)

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