Question

Transform the following functions to their natural exponential forms: (a) \(y=8^{3 t}\) (b) \(y=2(7)^{2 t}\) (c) \(y=5(5)^{t}\) \((d) y=2(15)^{4 t}\)

Short answer

(a) \(y = e^{3t \ln 8}\), (b) \(y = 2 e^{2t \ln 7}\), (c) \(y = 5 e^{t \ln 5}\), (d) \(y = 2 e^{4t \ln 15}\).

Step-by-step solution

Understanding the Problem

We want to express each given function in terms of the natural exponential function, which uses the base of natural logarithms, \(e\). The general strategy is to rewrite each expression by using the property of exponents that allows transforming any exponential function to a base \(e\).

Rewrite with Base \(e\) for (a)

For the function \(y=8^{3t}\), write it in terms of \(e\) using the identity: \(a^x = e^{x \ln a}\). We have:\[ y = e^{3t \ln 8} \].

Rewrite with Base \(e\) for (b)

For \(y=2(7)^{2t}\), use the identity again:We have:\[ y = 2 imes e^{2t \ln 7} \].

Rewrite with Base \(e\) for (c)

For the function \(y=5(5)^t\), we apply the transformation:\[ y = 5 imes e^{t \ln 5} \].

Rewrite with Base \(e\) for (d)

For \(y=2(15)^{4t}\), transform the expression to base \(e\):\[ y = 2 imes e^{4t \ln 15} \].

Key concepts

Transformation of Functions

Transformation of functions refers to changing the appearance or structure of a function, while preserving its properties and behaviors. This can include shifts, stretches, compressions, and rotations. In the context of converting between different exponential forms, transformation involves rewriting the function in a way that utilizes different bases, such as the natural exponential base, denoted as \(e\). By transforming functions, we can often simplify solving, analyzing, and modeling problems.In the exercises provided, the goal was to express different functions in terms of the natural exponential function. This transformation is achieved through understanding and applying the relationship between different exponential bases. Specifically, we leverage the identity \(a^x = e^{x \ln a}\), which allows any base to be expressed using the natural log base \(e\). This property is key when rewriting functions from any base \(a\) to the base \(e\). The transformation helps in scenarios where expressions involving \(e\) are more convenient for derivations or integrations due to their standard properties in calculus and other mathematical applications.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They have the general form \(y = a^x\), where \(a\) is the base and \(x\) is the exponent. These functions model growth and decay processes, like population dynamics, radioactive decay, and compound interest.Features of exponential functions:

• They exhibit rapid increase or decrease depending on the base.
• If the base \(a > 1\), the function represents exponential growth.
• If \(0 < a < 1\), the function signifies exponential decay.
• The graph of an exponential function is always a curve where the \(y\)-values change quickly compared to changes in \(x\).

In this exercise, exponential functions initially in forms such as \(8^{3t}\) or \(7^{2t}\) are converted so their bases change to \(e\), the natural base. This allows for easier manipulation using natural logarithms and understanding in advanced calculus applications, which often prefer the properties that come with the natural exponential base.

Base e Conversion

Converting any exponential expression to base \(e\) is a nuanced yet fascinating mathematical transformation. Base \(e\) (natural exponential base) approximately equals 2.71828 and serves as the foundation for natural logarithms. Transforming a function to base \(e\) involves changing its structure to one that utilizes natural logarithms, making it more compatible with various mathematical operations.To transform a function into a base \(e\) format, we use the key identity: \(a^x = e^{x \ln a}\). This identity relies on understanding that any base can be expressed in terms of \(e\) and its natural log. This conversion process is particularly useful in calculus because the derivative of \(e^{x}\) is \(e^{x}\) itself. This maintains consistency and simplifies computations.For instance, converting \(8^{3t}\) to base \(e\), we calculate \(e^{3t \ln 8}\). This form is equivalent but allows using deeper mathematical techniques that benefit from exponential and logarithmic rules specific to \(e\). Such conversions are foundational in higher mathematics, given the inherent beauty and simplification afforded by base \(e\).