Problem 4 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

\(f(x)=\log _{x} x\)

Step-by-Step Solution

Answer

The function \( f(x) = \log_{x} x \) simplifies to 1.

Step 1: Identify the Function

The function given is \( f(x) = \log_{x} x \). This means that we need to understand how the logarithm with the base equal to its argument behaves.

Step 2: Understand the Logarithm Properties

A logarithmic function \( \log_{a} b \) returns the power to which the base \( a \) must be raised to yield \( b \). In this case, our function simplifies as we are looking at \( \log_{x} x \).

Step 3: Simplify the Logarithmic Expression

According to the properties of logarithms, \( \log_{a} a = 1 \) for any positive base \( a \).

Step 4: Conclusion

Since \( f(x) = \log_{x} x \) and by the property of logarithms \( \log_{a} a = 1 \), it follows that \( f(x) = 1 \). Thus, the function simplifies to a constant value.

Key Concepts

properties of logarithms

To truly grasp how to work with logarithmic functions, we need to delve into some properties of logarithms. A logarithm is the inverse operation of an exponentiation. This means if we have \( a^c = b \), the logarithmic form would be \log_a b = c. Here, 'a' is the base, 'b' is the argument, and 'c' is the exponent. There are several properties of logarithms that are fundamental:

• Product Property: \log_a (bc) = \log_a b + \log_a c
• Quotient Property: \log_a \left( \frac{b}{c}\right) = \log_a b - \log_a c
• Power Property: \log_a (b^c) = c \log_a b
• Change of Base Formula: \log_a b = \frac{\log_c b}{\log_c a}

Understanding these properties is crucial for simplifying expressions and solving logarithmic equations. They help us break down complex logarithmic expressions into simpler parts.

logarithmic simplification

Simplifying logarithmic expressions is an essential skill in working with logarithms. In the given problem, we have the function \( f(x) = \log_x x \). Let's use the properties of logarithms to simplify this. According to the property \log_a a = 1\, the logarithm of a number with its own base is always 1. This is because \a^1 = a \. Applying this to our function: \ f(x) = \log_x x\. Here, the base and the argument are the same (both are 'x'). Therefore, \log_x x = 1\. So, the entire expression simplifies to 1. Hence, \f(x) = 1 \ for any positive value of 'x'. This drastically simplifies what initially looks like a complex function, showcasing the power of understanding the properties of logarithms.

constant function

A constant function is a function that always returns the same value, no matter the input. In the context of our exercise, we have simplified the function \(f(x) = \log_x x \) to \(f(x) = 1 \). This means that for any input value of 'x' (as long as 'x' is positive), the output is always 1. This is the hallmark of a constant function. Constant functions are unique in that their graph is a horizontal line. In this case, the line will be at \(y=1 \). These functions have some interesting properties:

• Their slope is 0 because there is no change in the output value.
• They are very predictable and easy to work with in various mathematical contexts.
• The derivative of a constant function is always 0, indicating no rate of change.

Recognizing a constant function can streamline solving algebraic problems and understanding their relevance in broader mathematical applications.