Question

Use the inverse relations between \(\ln x\) and \(\exp (x),\) and the properties of \(\ln x,\) to prove the following properties. a. \(\exp (0)=1\) b. \(\exp (x-y)=\frac{\exp (x)}{\exp (y)}\) c. \((\exp (x))^{p}=\exp (p x), p\) rational

Short answer

Short Answer: To prove the given properties of the exponential function, we relied on the inverse relations between ln(x) and exp(x), as well as the known properties of the natural logarithm. In each case, we started with the function to be proved, applied the inverse relations, used logarithm properties, and then applied the inverse relations again to get the required result. As a result, we proved that exp(0)=1, exp(x-y) = (exp(x))/(exp(y)), and (exp(x))^p = exp(px) when p is rational.

Step-by-step solution

Start with the exponential of zero

Write the function that needs to be proved: \(\exp(0)\).

Use the inverse relation btween \(\ln(x)\) and \(\exp(x)\)

Apply the inverse relation of \(\ln(\exp(x)) = x\) to \(\exp(0)\): \(\ln(\exp(0)) = 0\).

Identify the value of \(\exp(0)\)

Since \(\ln(1)=0\), we have \(\ln(\exp(0)) = \ln(1)\). This means that \(\exp(0) = 1\). So we have proved that \(\exp(0) = 1\). #b) Prove \(\exp(x-y) = \frac{\exp(x)}{\exp(y)}\)

Start with the function to prove

Write the function that needs to be proved: \(\exp(x-y)\).

Use the inverse relation btween \(\ln(x)\) and \(\exp(x)\)

Apply the inverse relation of \(\ln(\exp(z)) = z\) to \(\exp(x-y)\): \(\ln(\exp(x-y)) = x-y\).

Use properties of logarithms

Apply the property \(\ln \frac{a}{b} = \ln a - \ln b\) to \(\ln(\exp(x-y))=x-y\). We get \(\ln(\frac{\exp(x)}{\exp(y)})=x-y\)

Use the inverse relation again

Apply the inverse relation of \(\exp(\ln(z)) = z\) to \(\ln(\frac{\exp(x)}{\exp(y)})=x-y\). We get \(\exp(x-y)=\frac{\exp(x)}{\exp(y)}\). So we have proved that \(\exp(x-y) = \frac{\exp(x)}{\exp(y)}\). #c) Prove \((\exp(x))^{p} = \exp(px)\), for \(p\) rational

Start with the function to prove

Write the function that needs to be proved: \((\exp(x))^{p}\).

Use the inverse relation btween \(\ln(x)\) and \(\exp(x)\)

Apply the inverse relation of \(\ln(\exp(z)) = z\) to \((\exp(x))^p\): \(\ln((\exp(x))^p) = px\).

Use properties of logarithms

Apply the property \(\ln a^{p} = p\ln a\) to \(\ln((\exp(x))^p)=px\). We get \(p\ln(\exp(x))=px\)

Use the relationship between logarithms and exponentials

By using the inverse relation again, we know that \(\ln(\exp(x)) = x\). So \(p\ln(\exp(x))= px\) becomes \(px = px\) which is always true.

Use the inverse relation again

Apply the inverse relation of \(\exp(\ln(z)) = z\) to \(\ln((\exp(x))^p) = px\). We get \((\exp(x))^p=\exp(px)\). So we have proved that \((\exp(x))^p = \exp(px)\) when p is rational.

Key concepts

Inverse Relations of Logarithms and Exponentials

Understanding the inverse relation between logarithms and exponentials is crucial in advanced mathematics. Essentially, the exponential function and logarithmic function are inverses of each other. This means that the natural logarithm function, denoted as \( \ln(x) \), is the inverse of the exponential function \( \exp(x) \).

For any real number \( x \), the equation \( \exp(\ln(x)) = x \) holds true. Similarly, for any real number \( y \), the equation \( \ln(\exp(y)) = y \) is valid. This concept allows us to switch between exponential and logarithmic forms, which is a foundational skill for solving a plethora of algebraic problems.

Exponential Function Properties

The exponential function, represented as \( \exp(x) \), has a set of properties that are derived from its definition. One of the fundamental properties is that \( \exp(0) = 1 \), which states that any base raised to the power of zero equals one. This property is not only intuitive but aligns with the rules of exponents.

Another property is the quotient rule: \( \exp(x-y)=\frac{\exp(x)}{\exp(y)} \). This rule can be understood in the context of subtracting exponents, which is equivalent to dividing the base raised to these exponents. These properties of the exponential function are pivotal for simplifying complex expressions in algebra and calculus.

Logarithmic Properties

Logarithms have properties that mirror those of exponents due to their inverse relationship. One such property is the logarithm of a quotient: \( \ln \frac{a}{b} = \ln a - \ln b \). This property is directly used in the proof of how an exponential function behaves with a difference in exponents.

The property that allows us to log an exponent of a number (\( \ln a^{p} = p\ln a \)) is another log property used in the proofs of the properties of the exponential function. Recognizing and applying logarithmic properties are essential in reducing complex expressions and solving equations involving exponents and logarithms.

Rational Exponents

Rational exponents express roots and powers simultaneously and can be seen as an extension of the property of integer exponents. The notation \( a^{p/q} \), where \( p \) and \( q \) are integers, represents the \( q \)-th root of \( a \) raised to the power of \( p \).

When dealing with exponential functions, the property \( (\exp(x))^{p} = \exp(px) \) for \( p \) rational is an application of this concept, where an exponent can be multiplied with a power to simplify expressions. This concept is extensively used in calculus, particularly in differentiation and integration where exponents frequently vary.