Problem 1 - Algebra and Trigonometry Real Mathematics, Real People

Question

The inverse function of the exponential function \(f(x)=a^{x}\) is called the _____ with base \(a\).

Step-by-Step Solution

Answer

The inverse function of the exponential function \(f(x)=a^{x}\) is the logarithmic function with base \(a\).

Step 1: Understand concept of inverse function

By definition, an inverse function is a function that 'undoes' the process of another function. It reverses the effect of the original function.

Step 2: Identify inverse of exponential function

In the case of an exponential function \(f(x) = a^{x}\), the inverse function is the one that reverses the process of exponentiating a number to the power x. The function that accomplishes this is the logarithmic function with base \(a\).

Key Concepts

exponential function

Exponential functions are a type of mathematical function where the variable is located in the exponent. It takes the form \(f(x) = a^{x}\). Here, \(a\) is a constant called the base, and \(x\) is the exponent or power. These functions are characterized by rapid growth or decay.

For example, if \(a > 1\), the function will grow exponentially as \(x\) increases. Conversely, if \(0 < a < 1\), the function will decay exponentially as \(x\) becomes larger.

Exponential functions are encountered in a variety of real-world situations, such as population growth, radioactive decay, and compound interest calculations. Understanding how they work is crucial for interpreting and predicting patterns in these contexts."},{