Problem 1 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Concept of Inverse Function

The inverse of a function 'f' is the function that reverses the 'operation' of the original function. In simpler terms, the inverse of a function will take the output of the function and gives back the original input.

Step 2: Identify the Inverse of Exponential Function

The inverse of an exponential function is known as the logarithmic function, it does exactly reverse of what an exponential function does.

Step 3: Formulate the Answer

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

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. The general form of an exponential function can be written as \( f(x) = a^x \), where \( a \) is the base, and \( x \) is the exponent.
This function is unique because of its rapid growth or decay, depending on the base value.

• If the base \( a \) is greater than 1, the function represents exponential growth.
• If \( 0 < a < 1 \), it represents exponential decay.

Exponential functions are widely used in various fields like finance for compound interest calculations, biology for population growth models, and physics for radioactive decay.
A special property of exponential functions is that they are continuous and smooth over their domain of all real numbers.

Logarithmic Function

The logarithmic function is essentially the inverse of the exponential function. If you have an exponential function like \( f(x) = a^x \), its inverse is the logarithmic function \( f^{-1}(x) = \log_a(x) \).
This means the logarithm of a number is the exponent to which the base must be raised to produce that number.

• For example, \( \log_2(8) = 3 \) because \( 2^3 = 8 \).
• The logarithmic function is defined only for positive real numbers.

Logarithms are particularly useful in situations where dealing with exponential scales, such as in measuring sound intensity in decibels, or in solving equations where the unknown appears as an exponent.
Remember, since logarithmic functions are inverses of exponential functions, they "undo" each other. If you apply a logarithm to an exponential function, or vice versa, you retrieve the original input.

Base

In the context of exponential and logarithmic functions, the base is the number itself that is raised to a power or used in the logarithmic operation. It is a critical parameter that influences the behavior of these functions.
For an exponential function \( f(x) = a^x \), the base \( a \) determines the rate of growth or decay:

• A higher base means faster growth as \( x \) increases for \( a > 1 \).
• A base between 0 and 1 means the function will decay as \( x \) increases.

In logarithmic functions, the base \( a \) is used to determine the scale of measurement:

• Common bases are \( a = 10 \) (common logarithm), \( a = e \) (natural logarithm), and \( a = 2 \), each useful in different fields.
• The base chosen for a logarithm affects how the results are interpreted and applied in problem-solving.

Choosing the appropriate base is critical for solving real-world problems effectively and requires understanding the properties of these mathematical functions.