Problem 124 - Precalculus

Question

How is the logarithmic function \(f(x)=\log _{b} x\) related to the exponential function \(g(x)=b^{x} ?\) What is the result of composing these two functions?

Step-by-Step Solution

Answer

The functions are inverses; both compositions return \(x\).

Step 1: Define the Functions

We are given two functions: the logarithmic function \(f(x) = \log_b x\) and the exponential function \(g(x) = b^x\). The base \(b\) is a positive number, and typically \(b > 1\).

Step 2: Understand Composition of Functions

The composition of two functions \(f\) and \(g\), denoted \(f(g(x))\), means applying \(g\) first and then \(f\). Similarly, \(g(f(x))\) means applying \(f\) first and then \(g\).

Step 3: Compose g and f as g(f(x))

Let's first find \(g(f(x))\). Substitute \(f(x) = \log_b x\) into \(g(x) = b^x\). Hence, \(g(f(x)) = b^{\log_b x}\). By the property of exponents and logarithms, \(b^{\log_b x} = x\).

Step 4: Compose f and g as f(g(x))

Next, find \(f(g(x))\). Substitute \(g(x) = b^x\) into \(f(x) = \log_b x\). Hence, \(f(g(x)) = \log_b(b^x)\). By the property of logarithms, \(\log_b(b^x) = x\).

Step 5: Conclude the Relationship

Both compositions \(g(f(x))\) and \(f(g(x))\) return \(x\). Therefore, the exponential function \(g(x)=b^x\) and the logarithmic function \(f(x)=\log _{b} x\) are inverse functions of each other.

Key Concepts

Exponential Function

The exponential function is a powerful mathematical concept characterized by expressing numbers using a base raised to the power of exponents. When we talk about exponential functions, they generally take the form \( g(x) = b^x \), where \( b \) is a positive constant known as the base, and \( x \) is the exponent.
Exponential functions are remarkable because of their constant proportional growth rate. In other words, the rate of change of the function is directly proportional to the current value of the function itself.

Some common features of exponential functions include:

• Their domain is all real numbers, meaning \( x \) can take any value.
• Their range is all positive real numbers, which means \( g(x) = b^x > 0 \) for any \( x \).
• When the base \( b > 1 \), the function is an increasing function, exhibiting exponential growth.
• If \( 0 < b < 1 \), the function describes exponential decay, gradually approaching zero as \( x \) increases.

Exponential functions are used extensively in sciences, finance, and engineering, particularly because they model phenomena such as population growth, radioactive decay, and interest compounding.

Composition of Functions

Composition of functions is a process that involves applying one function to the results of another. When dealing with the composition of functions, you might see notations like \( f(g(x)) \) or \( g(f(x)) \). This can be visualized as wrapping one function within another, allowing us to combine functions in sophisticated ways.
For example, if \( f(x) \) is a logarithmic function and \( g(x) \) is an exponential function, you can create new functions by composing these in different orders: \( g(f(x)) \) or \( f(g(x)) \).

There are key steps in understanding composition:

• Firstly, identify which function is being applied first. For \( f(g(x)) \), \( g(x) \) is applied before \( f \).
• Apply the inner function to your input value.
• Use the result from the inner function as the input for the outer function.

Composing functions can simplify complex calculations by transforming problems into a series of simpler steps. It often leads to remarkable results, such as in the case of exponential and logarithmic functions where \( g(f(x)) = x \) and \( f(g(x)) = x \), confirming these functions are inverses of each other.

Inverse Functions

Inverse functions are a fundamental concept in mathematics where two functions "undo" each other. In simple terms, if a function \( f \) takes an input \( x \) to an output \( y \), then its inverse function \( f^{-1} \) will take \( y \) back to \( x \). This characteristic is crucial in equations and problem-solving processes.
For the exponential function \( g(x) = b^x \) and its inverse, the logarithmic function \( f(x) = \log_b x \), the property of being inverses means:

• Applying \( g \) to \( x \) and then \( f \) to \( g(x) \) will return the original value \( x \).
• Similarly, applying \( f \) and then \( g \) also returns to the starting value: \( g(f(x)) = x \).

Inverse functions have various applications, notably in solving exponential equations where logarithms are used to isolate the variable in the exponent. Recognizing functions as inverses simplifies solving problems by providing methods to revert changes and undo operations neatly. Being able to switch smoothly between a function and its inverse expands your toolset for tackling a plethora of mathematical challenges.