Problem 17 - College Algebra with Modeling and Visualization

Question

Find the domain of \(f\) and write it in setbuilder or interval notation. $$f(x)=\log _{3}\left(4^{x}\right)$$

Step-by-Step Solution

Answer

The domain of the function is all real numbers, \((-\infty, \infty)\).

Step 1: Identify the Argument of the Logarithm

The function given is \(f(x) = \log_3(4^x)\). The argument in the logarithm is \(4^x\). For a logarithm to be defined, its argument must be positive. Thus, the condition \(4^x > 0\) needs to be satisfied.

Step 2: Solve the Inequality

Next, we solve the inequality \(4^x > 0\). Since \(4^x\) is an exponential function with base 4, which is positive, \(4^x\) is always positive for any real number \(x\). There are no restrictions on \(x\), therefore the inequality is satisfied for all real numbers.

Step 3: Express the Domain

Since the condition \(4^x > 0\) is satisfied for all real numbers, the domain of \(f(x)\) is all real numbers. In interval notation, this is expressed as \((-\infty, \infty)\). In set builder notation, it can be expressed as \(\{x \mid x \in \mathbb{R}\}\).

Key Concepts

Setbuilder notation

Setbuilder notation is a concise way of specifying a set by indicating the properties that its members must satisfy. In mathematics, it's a useful way to describe complex sets using conditions.

When we use setbuilder notation, we write it as \( \{x \mid \, \text{condition(s)} \} \), where \( x \) stands for any element within the set, and the statements after \( \mid \) describe the rule that \( x \) must follow.

• For example, expressing the domain of all real numbers in setbuilder notation is written as \( \{x \mid x \in \mathbb{R} \} \).
• The notation means "the set of all \( x \) such that \( x \) is an element of the real numbers \( \mathbb{R} \)."

This form of expression is pivotal in defining the domain or range of functions clearly and succinctly.

By using setbuilder notation, mathematicians can precisely communicate which numbers can serve as inputs or outputs for a given function.

Interval notation

Interval notation provides a shorthand method to describe a range of values for which a function is defined or valid. It's particularly useful in expressing domains and ranges without needing lengthy descriptions.

In interval notation:

• The symbol \( ( \) or \( ) \) means that the endpoint is not included in the interval (open interval).
• While \( [ \) or \( ] \) indicates that the endpoint is included in the interval (closed interval).

For example, the set of all real numbers is written in interval notation as \((-\infty, \, \infty)\). This implies that the domain includes all possible real numbers:

• "minus infinity" to "infinity" signifies that you are considering all numbers that stretch endlessly in both the negative and positive directions.
• Parentheses around infinity tell us that infinity is a concept rather than a number we can reach or include.

This notation is a quick, clean way to capture the entirety of a function's domain, emphasizing where the function performs valid evaluations.

Logarithmic functions

Logarithmic functions are the inverses of exponential functions. They are essential in many areas of mathematics and sciences.

A logarithm answers the question: "To what power must a base be raised, to obtain a given number?"

• If \( b^y = x \), then \( y = \log_b(x) \).
• Here, \(b\) is the base of the logarithm, \(y\) is the logarithm, and \(x\) is the argument.

One of the critical aspects is that for logarithms to be defined, their argument must always be positive. This is because no real number raised to another real number can yield a negative or zero as a result.

For the function \(f(x)=\log _{3}\left(4^{x}\right)\), the base is 3. The argument, \(4^x\), determines the values \(x\) can take.

• Since \(4^x>0\) for all real numbers \(x\), the domain of \(f(x)\) includes all real numbers.

Logarithmic functions are widely used for modeling growth curves, computing complex scales like the Richter scale, and for financial calculations involving compound interest.