Problem 399 - Precalculus

Question

Prove that \(b^{x}=e^{x \ln (b)}\) for positive \(b \neq 1\)

Step-by-Step Solution

Answer

The identity \(b^{x} = e^{x \ln (b)}\) is proven through the properties of logarithms and exponentials.

Step 1: Define the Problem

We need to prove the mathematical identity that relates the exponential form to a natural exponential form: \(b^{x} = e^{x \ln(b)}\). Here, \(b\) is a positive number not equal to 1, and \(x\) is any real number.

Step 2: Apply the Definition of Exponentials

According to the definition of exponentials, \(e^{\ln(b)} = b\). This implies that raising \(e\) to the power of \(\ln(b)\) gives \(b\).

Step 3: Rewrite the Base of the Exponential

Using the understanding from the previous step, we can express \(b^x\) in terms of the base \(e\): \(b^x = (e^{\ln(b)})^x\).

Step 4: Apply the Power Rule of Exponents

The power rule states that \((a^m)^n = a^{m \cdot n}\). Applying this to our expression gives:\((e^{\ln(b)})^x = e^{x \cdot \ln(b)}\).

Step 5: Combine Results

From the steps above, we have\(b^x = e^{x \ln(b)}\). Thus, we have proven the identity that \(b^{x} = e^{x \ln (b)}\).

Key Concepts

Understanding Exponentials

Exponentials are mathematical expressions involving a base raised to a power or exponent. The general form is given by \(a^n\), where \(a\) is the base and \(n\) is the exponent. In such expressions, the exponent signifies how many times the base is multiplied by itself. This is known as repeated multiplication.
For example, \(3^2 = 3 \times 3 = 9\). By increasing complexity, exponentials become useful in many areas, especially in growth-related problems like population growth, compound interest, and radioactive decay.
Exponential identities, such as \(b^x = e^{x \ln(b)}\), allow transformations between bases and make complex exponential expressions more manageable by expressing them in terms of familiar constants like \(e\), the base of natural logarithms, approximately equal to 2.718. Thus, understanding exponentials facilitates simplifying and solving equations across various scientific fields.

The Role of Natural Logarithms

Natural logarithms are the inverse functions of exponentials, specifically those with base \(e\). They are denoted as \(\ln(x)\). In simple terms, while exponentials deal with the power \(a\) to which a given number (the base) is raised, logarithms answer the question: "To what power must the base, \(e\), be raised to obtain \(x\)?"
For example, finding \(\ln(e)\), results in 1 because \(e^1 = e\). Natural logarithms have special properties that make them suitable for simplifying expressions involving exponentials.

• They turn multiplication into addition: \(\ln(xy) = \ln(x) + \ln(y)\).
• They convert division into subtraction: \(\ln(x/y) = \ln(x) - \ln(y)\).
• The logarithm of a power turns multiplication into scalar multiplication: \(\ln(x^y) = y \cdot ln(x)\).

Such properties are essential for unraveling complex exponential expressions and proving identities like \(b^x = e^{x \ln(b)}\), as they translate an exponential problem into a linear one.

Exploring the Power Rule of Exponents

The power rule of exponents is a fundamental principle in algebra, greatly simplifying the manipulation of exponential expressions. It states that \((a^m)^n = a^{m\cdot n}\). This rule is crucial when working with nested exponentials or transforming expressions to simplify and solve them efficiently.
In the context of proving \(b^x = e^{x \ln(b)}\), the power rule allows us to rewrite a complex expression, \((e^{\ln(b)})^x\), as \(e^{x \cdot \ln(b)}\). By applying this principle, you can see how an equation with nested exponents – often difficult to handle directly – is reduced to a simpler expression that can be solved with standard tools.
These transformations are vital in fields ranging from pure mathematics to physics and engineering, where exponential and logarithmic relationships frequently arise. Understanding how to apply the power rule can help solve equations more intuitively and efficiently.