Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

Yes, \(f(x)=0\) is in the range for \(x=1\).

Step 1: Understand the Problem

We need to determine if the value 0 is in the range of the function \(f(x) = \log(x)\). This means finding an \(x\) such that \(f(x) = 0\).

Step 2: Set the Equation

We set the equation \(f(x) = \log(x) = 0\). This means we need to find the value of \(x\) that fulfills this equation.

Step 3: Solve the Equation

We solve \(\log(x) = 0\). By properties of logarithms, \(\log(x) = 0\) implies that \(x = 10^0 = 1\).

Step 4: Verify the Solution

Substitute \(x = 1\) back into the function to verify: \(f(1) = \log(1) = 0\). The result confirms that the calculation is correct and \(f(x) = 0\) is in the range of \(f(x)\) for \(x = 1\).

Key Concepts

Function Range

The range of a function refers to all the output values it can produce. For the logarithmic function \(f(x) = \log(x)\), the output depends on the input values of \(x\). Understanding the range helps us know how far the values stretch on the y-axis.

• Logarithmic Functions: These functions only accept positive values of \(x\) and stretch along the x-axis towards infinity.
• Output Values: Since \(f(x) = \log(x)\) can produce both positive and negative outputs, its range is all real numbers.
• Example: When \(x=1\), \(f(x)=\log(1)=0\). Therefore, 0 is within the real numbers range of the function.

Recognizing the range is crucial for understanding what results are possible, and it helps identify solutions when working on any equation related to logarithmic functions.

Properties of Logarithms

Logarithms have specific properties that simplify the solving of equations involving them. These properties are essential tools in both practical and theoretical mathematics.

• Identity Property: The property \(\log_b(b) = 1\) helps when working with multiples of the base.
• Zero Property: \(\log_b(1) = 0\)
• Power Rule: Allows for simplifying exponents, \(\log_b(x^y) = y \cdot \log_b(x)\).

Using the zero property, we know that \(\log_b(1) = 0\) provides a simple solution when determining against which values to verify results. The zero property is how we easily solved that for \(f(x) = \log(x)\), the equation gives zero when \(x\) equals one.

Solving Logarithmic Equations

Solving logarithmic equations involves applying rules and properties wisely. Recognizing which properties to utilize is critical in finding correct solutions.
When tasked with solving equations like \(\log(x) = 0\), the zero property directly applies. Here’s a step-by-step approach:

• Equation Setup: Identify what the equation \(\log(x) = 0\) suggests.
• Property Application: Apply the zero property (\(\log_b(1) = 0\)) implying that the solution for the equation should satisfy \(x=1\).
• Verification: Always verify results. Substitute back into the function: \(f(1) = \log(1) = 0\).

These steps ensure that you accurately solve logarithmic equations and confirm that potential solutions fit within the equation parameters. Using properties correctly makes the process straightforward, allowing for a clear and confident approach to solving.