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Chapter 5: Problem 114

Find the domain of the function. \(f(x)=\log (3 x-4)\)

Short Answer

Expert verified
The domain of \(f(x) = \log(3x-4)\) is \(x > \frac{4}{3}\).

Step by step solution

01

- Understanding Logarithmic Functions

Recall that the logarithmic function \(\log (y)\) is only defined when \(y > 0\). Here, \(y = 3x - 4\). Therefore, the argument of the logarithm, \(3x - 4\), must be greater than zero.
02

- Setting Up the Inequality

Set up the inequality \(3x - 4 > 0\) to find the values of \(x\) for which the function \(f(x) = \log (3x - 4)\) is defined.
03

- Solving the Inequality

Solve the inequality \(3x - 4 > 0\): \[3x - 4 > 0 \ 3x > 4 \ x > \frac{4}{3}\]
04

- Writing the Domain

Since \(x\) must be greater than \(\frac{4}{3}\) for \(f(x) = \log (3x - 4)\) to be defined, the domain of the function is \(x > \frac{4}{3}\).

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Key Concepts

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

Logarithmic functions
A logarithmic function is an inverse of an exponential function. The general form is \(\text{log}_b(y)\) where \(b\) is the base. It is only defined for positive arguments. This means \(y > 0\). The logarithm \(\log_b(y)\) answers the question: 'To which power must we raise \(b\) to get \(y\)?' In our exercise, we have \(\text{log}(3x-4)\). For this function to be defined, the input \(3x-4\) must be positive.
Inequalities
Inequalities involve comparing two quantities. Instead of stating that two quantities are equal, inequalities tell us how one quantity is related to another. In the case of \(\text{log}(3x-4)\), we need to ensure that \(3x-4 > 0\). This is an inequality. It tells us that we need a specific range of values for \(x\) to make \(3x-4\) positive. Here, \(3x-4\) needs to be greater than zero to ensure the function is defined.
Solving inequalities
To solve the inequality \((3x - 4) > 0\), we follow these steps:
1. Add 4 to both sides: \(3x - 4 + 4 > 0 + 4\) which simplifies to \(3x > 4\).
2. Divide both sides by 3: \(3x/3 > 4/3\) which simplifies to \(x > \frac{4}{3}\).
This means \(x\) must be greater than \(\frac{4}{3}\). So, the domain of \(f(x) = \log(3x - 4)\) is all \(x\) such that \(x > \frac{4}{3}\).

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