determine the domain of the function y=log(subscript/little x)(8-x)

so x^y = 8-x
how do I find the domain?

To determine the domain of the function y = logₓ(8 - x), you first need to understand the restrictions on the logarithmic function.

In general, the domain of a logarithmic function is determined by two main factors:
1. The base (in this case, x) should be positive and not equal to 1.
2. The expression inside the logarithm (8 - x) should be positive.

Let's break down these restrictions:

1. The base (x) should be positive and not equal to 1:
Since x is the base of the logarithm, it must be greater than 0 (positive). Additionally, it cannot be equal to 1 since logarithms with a base of 1 are undefined.

Therefore, the domain for this function is x > 0 and x ≠ 1.

2. The expression inside the logarithm (8 - x) should be positive:
In this case, the expression inside the logarithm is (8 - x). For the logarithmic function to be defined, this expression must be greater than 0.

Setting (8 - x) > 0 and solving for x:
8 - x > 0
-x > -8
x < 8

Therefore, another restriction for the domain is x < 8.

To find the overall domain, you need to consider both restrictions together. Combining the conditions x > 0 and x ≠ 1 with x < 8, the final domain for the function y = logₓ(8 - x) is:

0 < x < 8, and x ≠ 1