Problem 37 - Contemporary Precalculus

Question

Write the rule of the function in the form \(\left.f(x)=P e^{k x} . \text { (See the discussion and box after Example } 11 .\right)\) $$f(x)=4\left(25^{x}\right)$$

Step-by-Step Solution

Answer

Question: Rewrite the function \(f(x)=4\left(25^x\right)\) in the form \(f(x) = Pe^{kx}\). Answer: The function can be rewritten as \(f(x) = 4e^{x\ln{25}}\), where P = 4 and k = \(\ln{25}\).

Step 1: Rewrite the given function with a base \(e\) exponent

To rewrite the function with a base \(e\) exponent, we'll need to use the property of exponents: \(a^{bx}=(e^{(\ln{a})})^{bx}=e^{\ln{a^b}x}=e^{b\ln{a}x}\). This allows us to rewrite the given function \(f(x)=4\left(25^x\right)\) as follows: $$f(x) = 4\left(e^{\ln{25^x}}\right)$$

Step 2: Write the function in the form \(f(x) = Pe^{kx}\)

Now, we will apply the exponent properties to simplify and write the function in the desired form. We'll replace \(P\) and \(k\) values accordingly: $$f(x) = 4\left(e^{x\ln{25}}\right) = (4)e^{(x\ln 25)}$$ Here, \(P=4\) and \(k=\ln{25}\). Therefore, the function can be written in the form \(f(x)=Pe^{kx}\) as: $$f(x) = 4e^{x\ln{25}}$$

Key Concepts

Exponent Properties

Understanding exponent properties is crucial for simplifying and manipulating expressions with exponents. The fundamental properties include the product of powers, the power of a power, and the power of a product.

For example, the product of powers property states that when you multiply two exponents with the same base, you can add the exponents: \( a^m \cdot a^n = a^{m+n} \). Similarly, the power of a power property allows you to multiply the exponents when one exponent is raised to another: \( (a^m)^n = a^{m \cdot n} \). Lastly, the power of a product property demonstrates that when a product is raised to an exponent, every factor of the product is raised to that exponent: \( (ab)^m = a^m \cdot b^m \).

These properties become particularly handy when dealing with more complex expressions, allowing us to rewrite them in various forms, especially when transforming to base \( e \) exponents which is common in calculus and higher-level mathematics.

Natural Logarithm

The natural logarithm is denoted by \( \ln \) and is the inverse operation of taking an exponent of \( e \) where \( e \) is the base of the natural logarithm. The natural logarithm of a number \( x \) returns the exponent to which \( e \) would have to be raised to equal \( x \) - that is \( \ln(x) \), such that \( e^{\ln(x)} = x \).

In the provided exercise, we use the natural logarithm to rewrite the base 25 as an exponent of \( e \) by taking \( \ln(25) \). This step is crucial because it leverages the natural logarithm’s properties of logarithmic and exponential functions being inverses of each other to transform expressions into a form involving \( e \) which is more amenable to calculus operations.

Transformations of Functions

Transformations of functions are operations that alter the graph of a function in various ways. This can include shifting, stretching, compressing, and reflecting the graph. For example, adding a constant to the function \( f(x) \) shifts the graph vertically, while multiplying \( f(x) \) by a constant can stretch or compress it vertically.

When we rewrite the function \( f(x)=4(25^x) \) as \( f(x) = 4e^{x\ln{25}} \) using the properties of \( e \) and natural logarithms, we are essentially transforming the function into an equivalent form that exhibits exponential growth. The coefficient 4 is a vertical stretch of the function, and \( \ln{25} \) affects the rate of growth, making it possible to express the function in terms of \( e \) which is the preferred base for many mathematical applications.

Base \( e \) Exponents

Exponents with base \( e \) (Euler's number, approximately equal to 2.71828) are pervasive in mathematics, especially in the domain of continuous growth and decay models. Euler's number has unique properties that make it a natural choice for describing processes of natural growth and continuous compounding.

In calculus, the derivative of \( e^x \) is itself, \( e^x \) - a property not shared by other bases. In the exercise, by expressing the function with a base \( e \) exponent, we align with this powerful property. We reframe \( 25^x \) as \( e^{x\ln{25}} \) to tap into the advantageous calculus tools available for base \( e \) exponential functions, setting the stage for deeper analysis and easier manipulation within various mathematical fields.