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Chapter 4: Problem 7

Write the domain in interval notation. $$ f(x)=\log _{5}(5-3 x) $$

Short Answer

Expert verified
The domain is \( (-\infty, \frac{5}{3}) \).

Step by step solution

01

Identify the domain constraint for a logarithmic function

The argument of the logarithmic function must be greater than zero. Thus, set the inequality: \[5 - 3x > 0\]
02

Solve the inequality

Isolate the variable x by subtracting 5 from both sides of the inequality: \[ -3x > -5 \]Then, divide both sides by -3, remembering to reverse the inequality sign: \[ x < \frac{5}{3} \]
03

Write the domain in interval notation

The inequality \( x < \frac{5}{3} \) means that x can take any value less than \(\frac{5}{3}\). Therefore, the domain in interval notation is: \( (-\infty, \frac{5}{3}) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. In simpler terms, if you have an equation in the form of \(y = \log_b(x)\), then it means that \(b^y = x\). The base of the logarithm, represented by b, must be a positive number and not equal to 1. Here, we are dealing with the function \(f(x) = \log_5(5 - 3x)\). Logarithmic functions have certain rules you must follow:
  • The argument (inside the log) must be a positive number. This means \(5 - 3x > 0\) must hold true.
  • The logarithms of negative numbers or zero are not defined.
  • The base must always be greater than zero and not equal to one.
Understanding these rules enables you to find the domain of logarithmic functions, ensuring their arguments always remain positive.
Inequality Solving
To solve the inequality \(5 - 3x > 0\), you need to manipulate the inequality to isolate x. Follow these steps:
  • Subtract 5 from both sides: \(-3x > -5\).
  • Since we are dividing by a negative number, reverse the inequality sign: \(x < \frac{5}{3}\).
This inequality shows that x must be less than \(\frac{5}{3}\). The solution tells us that all x-values satisfying this inequality are valid for the function. Always remember to reverse the inequality sign when multiplying or dividing by negative numbers to maintain the correctness of the solution.
Interval Notation
Interval notation is a concise way of expressing the set of numbers that satisfy an inequality. For our inequality \(x < \frac{5}{3}\), we write the domain in interval notation. Here's how:
  • Start with the smallest number in the range, which is \(-\text{infinity}\).
  • End with the largest number that satisfies the inequality, in our case, \(\frac{5}{3}\).
Thus, the domain of \(f(x) = \log_5(5 - 3x)\) in interval notation is written as \((- \infty, \frac{5}{3})\). This shows that x can be any value less than \(\frac{5}{3}\), but never equals or exceeds it. Interval notation efficiently communicates the permissible values of x for the given function.

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