Problem 60 - College Algebra with Corequisite Support

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, for \( x = 1 \), \( f(x)=0 \).

Step 1: Understand the Problem

We need to find out if the equation \( f(x) = \log(x) \) can output a value of 0. We are searching for such an \( x \) where the logarithm of \( x \) is zero.

Step 2: Recall Logarithmic Property

Recall that the logarithmic function \( \log(x) \) is defined such that \( \log_b(a) = c \) means \( b^c = a \). In our problem, we are dealing with the natural logarithm, so the base is \( 10 \). Thus, \( \log(x) = 0 \) means \( 10^0 = x \).

Step 3: Solve for x

Set up the equation using the property: \( \log(10^0) = 0 \). This implies \( x = 10^0 = 1 \). So, \( x = 1 \) is the value we are looking for.

Step 4: Verify the Solution

Substitute \( x = 1 \) into the function: \( f(x) = \log(1) \). Since \( 10^0 = 1 \), thus \( \log(1) = 0 \). This verifies that when \( x = 1 \), \( f(x) = 0 \) holds true.

Key Concepts

Natural Logarithm

A natural logarithm is a special kind of logarithm where the base is the number \( e \), which is approximately equal to 2.71828. In mathematical notation, we represent this as \( \ln(x) \), indicating the logarithm with a base of \( e \). Natural logarithms are extensively used in calculus and other areas of mathematics because they have properties that simplify differentiation and integration.
One of the most important features of natural logarithms is how they can transform multiplicative relationships into additive ones. This is extremely useful in various fields such as economics and biology, where exponential growth models are common. The inverse of the natural logarithm is the exponential function, which makes these concepts deeply interconnected.
When working with natural logarithms, remember:

• \( \ln(e) = 1 \) because \( e^1 = e \).
• \( \ln(1) = 0 \) since any number raised to the power of zero is one.

This is why in the solved exercise, \( \log(1) = 0 \) if we consider \( \log \) to be \( \ln \), meaning that the base could potentially be \( e \). Understanding these core properties allows you to manipulate and solve equations involving natural logarithms effectively.

Exponential Equations

Exponential equations involve variables in the exponent and are important because they model real-world phenomena like population growth and radioactive decay. An equation is said to be exponential if it takes the form \( a^x = b \), where \( a \) is a constant and \( x \) is the variable.
To solve an exponential equation, you often use logarithms. By taking the logarithm of both sides of the equation, you can "move" the variable down from the exponent, making it easier to solve. For example, solving \( 2^x = 8 \) involves taking the logarithm of each side:

• \( x \ln(2) = \ln(8) \)
• \( x = \frac{\ln(8)}{\ln(2)} \)

In many cases, exponential equations can be simplified using properties of logarithms.
These transformations are crucial in mathematical modeling and problem-solving. Always ensure the base of the logarithm matches the base of the exponential. As shown in the original problem, understanding logarithmic identities can allow you to quickly determine solutions for specific values.

Logarithmic Properties

Understanding logarithmic properties is essential for manipulating and solving complex logarithmic equations. Here are some of the key properties that can help simplify your calculations:

• Product Property: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Property: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Property: \( \log_b(x^n) = n \cdot \log_b(x) \)
• Change of Base Formula: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), where \( k \) is any valid base.

These properties simplify the process of solving logarithmic equations by transforming them in a way that exposes the variable more clearly.
In the provided problem, the key property used was the fact that \( \log(1) = 0 \), stating that any logarithm of 1, regardless of the base, is always 0. Being adept at using these properties helps not only in solving equations but also in verifying solutions, as was done in the original exercise when substituting \( x = 1 \) into the function to confirm \( f(x) = 0 \).