Conditions on the Base and the Argument — Cheat sheet

The three conditions

\log_a x \quad\text{needs}\quad a > 0, \quad a \ne 1, \quad x > 0

The value has no restriction. It may be negative, fractional, or zero.

Condition	Reason
$a > 0$	a negative base breaks between whole powers: $(-4)^{1/2}$ is not real
$a \ne 1$	every power of $1$ is $1$ , so $1^y = x$ has no single answer
$x > 0$	a positive base to any real power is positive, never $0$ or negative

One inequality per condition, then keep what they all allow.

Both base conditions apply, and the argument condition applies as well. Three inequalities.

\log_{x-1}(6 - 2x): \quad x > 1, \quad x \ne 2, \quad x < 3 \;\Rightarrow\; 1 < x < 3, ; x \ne 2

$x = 3$ is not allowed in $\log_5(x - 3)$ . The argument must be strictly positive, not zero.
A negative $x$ is fine if the argument still comes out positive, as in $\log_2(x^2 - 9)$ at $x = -4$ .
Not every logarithm restricts the domain. Check the argument before assuming it does.