Problem 59 - College Algebra

Question

Is \(x=0\) in the domain of the function \(f(x)=\log (x)\) ? If so, what is the value of the function when \(x=0 ?\) Verify the result.

Step-by-Step Solution

Answer

No, \( x = 0 \) is not in the domain. \( f(0) \) is undefined.

Step 1: Understand the Function

The function given is \( f(x) = \log(x) \), which is a logarithmic function. The logarithm is only defined for values greater than zero. This means we need to consider this in order to determine if \( x = 0 \) is part of the domain of \( f(x) \).

Step 2: Identify the Domain

The function \( f(x) = \log(x) \) is only defined for \( x > 0 \). This is because the logarithm of a non-positive number (like zero or a negative number) is undefined in real numbers.

Step 3: Check if \( x = 0 \) is in the Domain

Since the domain of \( f(x) = \log(x) \) is exclusively \( x > 0 \), \( x = 0 \) is not in the domain of this function. The logarithm function does not allow zero as an input.

Step 4: Determine the Value of the Function at \( x = 0 \)

Since \( x = 0 \) is not in the domain of the function, there is no real value for \( f(0) \). It is considered undefined.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial for determining the set of inputs that a function can accept. In mathematical terms, the domain is the complete set of possible values of the independent variable, usually represented as "x".

For logarithmic functions, such as \( f(x) = \log(x) \), the domain consists solely of positive real numbers. This is because the logarithm is only defined for values within this range.

In simple words, you cannot take the logarithm of zero or a negative number within the real number system. Therefore, when examining whether \( x = 0 \) is part of the domain of \( f(x) = \log(x) \), we find that the answer is no, as zero is not a positive number.

Undefined Values

Undefined values in mathematics occur when an operation or function does not result in a meaningful value. For instance, in the case of logarithmic functions like \( f(x) = \log(x) \), attempting to evaluate the function at \( x = 0 \) leads to an undefined scenario.

Why exactly is \( x = 0 \) not allowed? Because the logarithmic function is only defined for positive numbers in the context of real numbers.

Since \( x = 0 \) does not satisfy this condition (it is neither positive nor negative), the function cannot output a real value. Thus, any attempt to calculate \( \log(0) \) results in an undefined mathematical expression.

Logarithm Properties

Logarithms have unique properties and rules that govern their behavior. These properties are essential for performing operations involving logarithms, such as simplification or solving equations.

Some of the fundamental properties of logarithms include:

• \( \log_b(1) = 0 \): The log of 1 to any base is always 0, because any number raised to the power of 0 is 1.
• \( \log_b(b) = 1 \): The log of a base raised to 1 is the exponent itself, which is why \( \log_b (b) \) equals 1.
• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)

However, these properties cannot define a value for \( \log(x) \) when \( x \leq 0 \). This is why understanding these properties helps, but they effectively reinforce the need for \( x \) to be positive when dealing with any logarithmic function.