Problem 71 - Calculus Early Transcendentals

Question

Properties of \(e^{x}\) Use the inverse relations between \(\ln x\) and \(e^{x}\) and the properties of \(\ln x\) to prove the following properties. a. \(e^{x-y}=\frac{e^{x}}{e^{y}}\) b. \(\left(e^{x}\right)^{y}=e^{x y}\)

Step-by-Step Solution

Answer

Question: Prove the following properties of the exponential function \(e^x\) using the inverse relationships between \(\ln x\) and \(e^x\) and the properties of \(\ln x\): 1. \(e^{x-y}=\frac{e^{x}}{e^{y}}\) 2. \(\left(e^{x}\right)^{y}=e^{x y}\) Answer: Using the properties of logarithms and the inverse relationships between \(\ln x\) and \(e^x\), we have proven that \(e^{x-y}=\frac{e^{x}}{e^{y}}\) and \(\left(e^{x}\right)^{y}=e^{x y}\).

Step 1: Property 1: Proving \(e^{x-y}=\frac{e^{x}}{e^{y}}\)

Step 1: Rewrite the expression with logarithms. To prove \(e^{x-y}=\frac{e^{x}}{e^{y}}\), we should first rewrite the expression using logarithms since we are given properties of \(\ln x\). For that, we take the natural logarithm (base \(e\)) of both sides of the equation: $$\ln\left(e^{x-y}\right) = \ln\left(\frac{e^{x}}{e^{y}}\right)$$ Step 2: Apply properties of logarithms. We will now apply the properties of logarithms to simplify both sides of the equation: Property 1 (Power rule): \(\ln(a^b) = b\cdot\ln(a)\) Property 2 (Quotient rule): \(\ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) Applying the properties to our equation: $$ (x-y)\cdot\ln(e) = \ln(e^x) - \ln(e^y) $$ $$ (x-y)\cdot\ln(e) = x\cdot\ln(e) - y\cdot\ln(e) $$ Step 3: Simplify the equation. Since \(\ln(e) = 1\) (because \(e\) is the base of the natural logarithm), we can simplify the equation to: $$ x - y = x - y $$ Since both sides of the equation are equal, it proves that the property \(e^{x-y}=\frac{e^{x}}{e^{y}}\) is true.

Step 2: Property 2: Proving \(\left(e^{x}\right)^{y}=e^{x y}\)

Step 1: Rewrite the expression with logarithms. To prove \(\left(e^{x}\right)^{y}=e^{x y}\), we should first rewrite the expression using logarithms. For that, we take the natural logarithm of both sides of the equation: $$\ln\left(\left(e^{x}\right)^{y}\right) = \ln\left(e^{xy}\right)$$ Step 2: Apply properties of logarithms. We will now apply the properties of logarithms to simplify both sides of the equation: Property 1 (Power rule): \(\ln(a^b) = b\cdot\ln(a)\) Applying the properties to our equation: $$ y \cdot \ln(e^x) = x y \cdot \ln(e) $$ Step 3: Simplify the equation. Since \(\ln(e) = 1\), we can simplify the equation to: $$ y \cdot x = x y $$ Since both sides of the equation are equal, it proves that the property \(\left(e^{x}\right)^{y}=e^{x y}\) is true. In conclusion, using the inverse relationships between \(\ln x\) and \(e^x\), we have proven both given properties: \(e^{x-y}=\frac{e^{x}}{e^{y}}\) and \(\left(e^{x}\right)^{y}=e^{x y}\).

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is crucial when manipulating exponential and logarithmic expressions.
Logarithms are essentially the inverse operation of exponentiation, making them incredibly useful in simplifying complex expressions and solving equations.

Here are a few key properties of logarithms:

• **Product Rule:** The logarithm of a product is the sum of the logarithms of the factors: \( \ln(ab) = \ln(a) + \ln(b) \).
• **Quotient Rule:** The logarithm of a quotient is the difference of the logarithms: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• **Power Rule:** The logarithm of a power is the exponent times the logarithm of the base: \( \ln(a^b) = b \cdot \ln(a) \).

These rules make it easier to handle expressions where variables are in the exponent, as seen in the proving of properties like \( e^{x-y} = \frac{e^x}{e^y} \).
By rewriting such expressions in terms of logarithms, as done in the problem, we can break down the logic of the expression into manageable and provable steps.

Algebraic Proofs

Algebraic proofs allow us to establish the truth of mathematical statements through logical reasoning and mathematical operations.
In the context of exponents and logarithms, these proofs often require the manipulation of expressions using both algebraic and logarithmic properties.

An essential approach in algebraic proofs is the re-expression of terms to reveal relationships or simplify calculations.In the given exercise, the algebraic proofs rely significantly on understanding and applying properties of logarithms and exponentials correctly.

• By employing the Quotient Rule and Power Rule of logarithms, we can take complicated exponential expressions and rewrite them in terms of simpler logarithmic expressions.
• Then, using properties such as \( \ln(e) = 1 \), we can proceed to simplify to verify that both sides of the equation are equal, thus confirming the original statement is true.

This precision and adherence to properties make algebraic proofs a powerful tool for verifying mathematical truths.

Inverse Functions

Inverse functions reverse the operation of a given function, which is why they are important in solving equations involving logarithms and exponentials.
When we look at the functions \( e^x \) and \( \ln x \), they are inverses of each other.

This means applying one function, followed by the inverse, returns you to the original value:

• The function \( \ln(e^x) = x \).
• Similarly, \( e^{\ln x} = x \).

Because of this inverse relationship, when proving properties involving exponential and logarithmic expressions, we can often exploit these inverses.
In our exercise, we used \( \ln x \) to bring the exponential expressions into a form where logarithmic properties could be applied, thereby simplifying and proving the original expressions.
Understanding inverse functions helps us see the deeper symmetry between logarithms and exponential growth or decay processes.