To determine the domain of the function

$f(x) = \log_3(5 - 4x),$

we must ensure that the argument of the logarithm is positive, as the logarithmic function is only defined for positive values of its argument.

Step 1: Set the argument greater than 0

The argument of the logarithm is $5 - 4x$. Therefore, we solve the inequality:

$5 - 4x > 0.$

Step 2: Solve the inequality

$-4x > -5 \quad \text{(Subtract 5 from both sides)}.$ $x < \frac{5}{4} \quad \text{(Divide by -4 and reverse the inequality)}.$

Step 3: Write the domain

The domain of $f(x)$ is all values of $x$ such that:

$x < \frac{5}{4}.$

In interval notation, the domain is:

$(-\infty, \frac{5}{4}).$

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Would you like more details on this calculation or how to graph the function to verify the domain?

Here are 5 related questions to expand on this concept:

1. What is the range of $f(x) = \log_3(5 - 4x)$?
2. How does the base of the logarithm affect the function's growth?
3. What happens to the domain if the function is $f(x) = \ln(5 - 4x)$ instead?
4. Can we shift the logarithmic function to change its domain?
5. How would the domain change if $f(x) = \log_3(-5 + 4x)$?

Tip: Always check the argument of a logarithmic function to ensure it's strictly positive.