Problem 98 - Algebra and Trigonometry

Question

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 7 x^{-1}} $$

Step-by-Step Solution

Answer

The simplified form of \(e^{\ln 7 x^{-1}}\) is \(7x^{-1}\), which can also be written as \(\frac{7}{x}\).

Step 1: Recognize the relationship between 'e' and 'ln'

The exponential function \(e^x\) and the natural logarithm function \(\ln(x)\) are inverse functions. This means that \(e^{\ln x} = x\), for \(x > 0\).

Step 2: Apply the inverse property

In this case, \(x = 7x^{-1}\). So by the property of logarithms and exponential defines earlier, we find that \(e^{\ln(7x^{-1})} = 7x^{-1}\).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In this context, the base is often the irrational number \( e \) which is approximately equal to 2.718. The expression in an exponential function can be written as \( e^x \) where \( e \) is the base and \( x \) is the exponent.

Exponential functions are widely used in various fields such as biology, economics, and physics, often modeling growth or decay processes. They have a key property where the rate of change of the function is proportional to its current value. This makes them incredibly powerful in defining natural phenomena and processes.

Some basic properties of exponential functions include:

• \( e^{x+y} = e^x \, e^y \)
• \( \frac{e^x}{e^y} = e^{x-y} \)
• The derivative of \( e^x \) with respect to \( x \) is \( e^x \), making it unique.

Understanding these properties helps in manipulating and solving exponential expressions.

Inverse Functions

Inverse functions essentially "undo" each other. For two functions to be inverses, the output of one function, when passed through the other, should revert to the original input. This can be expressed as \(f(g(x)) = x\) and \(g(f(x)) = x\).

In the realm of logarithmic and exponential functions, the natural logarithm \( \ln(x) \) and exponential function \( e^x \) are classical examples of inverse functions. If you apply \( e \) to the \( \ln(x) \) of a number, you get the original number back: \( e^{\ln(x)} = x \). Similarly, taking the natural logarithm of \( e^x \) will give \( x \): \( \ln(e^x) = x \).

This inverse relationship is incredibly useful for simplifying complex algebraic expressions involving \( e \) and \( \ln() \). It allows one to easily solve for \( x \) when given equations that combine these two functions.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm with the base \( e \). It is the inverse function of the exponential function \( e^x \). In simple terms, if \( y = e^x \), then \( x = \ln(y) \).

It has several important properties that make it useful in calculus and algebra:

• \( \ln(1) = 0 \)
• \( \ln(e) = 1 \)
• \( \ln(a \, b) = \ln(a) + \ln(b) \)
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• \( \ln(a^b) = b \ln(a) \)

The natural logarithm is peculiar in its growth pattern; it grows slower than any linear polynomial despite initially growing faster at the origin. Its significance in calculus is due to its derivative; when you differentiate \( \ln(x) \), you get \( \frac{1}{x} \), making it invaluable in integration and solving differential equations.

Simplification of Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. This often means applying mathematical properties and identities to combine or reduce terms.

In the case of exponential expressions involving the natural logarithm, simplification takes advantage of their inverse relationship. For example, the task might involve reducing something like \( e^{\ln(x)} \), which can be simplified to \( x \) since \( e \) and \( \ln \) cancel each other out.

Let’s consider the expression \( e^{\ln(7x^{-1})} \). To simplify this, utilize the property where \( e^{\ln(a)} = a \). Thus, \( e^{\ln(7x^{-1})} \) simplifies directly to \( 7x^{-1} \).

Simplifying complex expressions makes calculations easier and is necessary for solving equations efficiently. Understanding the underlying relationships between functions greatly assists in quickly reaching the simplest form of an expression.