Problem 76 - Precalculus

Question

Determine whether each statement is true or false. $$e^{\ln x^{2}}=x^{2}$$

Step-by-Step Solution

Answer

True; the expression simplifies to \( x^2 \).

Step 1: Understand the given expression

We start by examining the given expression \( e^{\ln x^{2}} \). The expression involves the natural logarithm (ln) and the exponential function (\(e^x\)). The task is to simplify or evaluate this expression to see if it matches \( x^2 \).

Step 2: Simplify using properties of logarithms

Recall the identity \( \ln x^{n} = n \ln x \). Apply this property to simplify \( \ln x^{2} \):\[ \ln x^{2} = 2 \ln x \]

Step 3: Apply the exponential and logarithm inverse property

The exponential function \( e^x \) and the natural logarithm function \( \ln x \) are inverses of each other. Therefore, \( e^{\ln y} = y \). Using this property, we can simplify \( e^{2 \ln x} \):\[ e^{2 \ln x} = (e^{\ln x})^2 = x^2 \]

Step 4: Verify the simplification

By simplifying the expression \( e^{\ln x^{2}} = e^{2 \ln x} \, \text{to} \, x^{2} \), we observe that the original statement holds true. The exponential and logarithm identities used confirm the equation.

Key Concepts

Natural logarithm properties

The natural logarithm, abbreviated as \( \ln \), is a powerful mathematical tool used to solve problems involving exponential growth and decay. It is the inverse of the exponential function \( e^x \). For any positive number \( x \), the natural logarithm defines that number as the power to which \( e \) (Euler's number, approximately 2.718) must be raised to equal \( x \). This could be expressed mathematically as \( e^{\ln x} = x \). The base \( e \) is chosen because it makes the calculus of logarithmic functions particularly elegant in terms of derivatives and integrals.

The properties of natural logarithms make them useful for simplifying expressions:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln(a/b) = \ln a - \ln b \)
• \( \ln a^n = n \ln a \)

Understanding these properties allows one to manipulate and simplify logarithmic expressions much easier. Together, they form the backbone of logarithmic calculations.

Exponential function properties

Exponential functions, which include expressions written as \( e^x \), are widely used to model growth and decay processes. A defining characteristic of the exponential function is that the rate of change is directly proportional to the value of the function. In simpler terms, the function grows faster as it gets larger.

Some important properties of exponential functions include:

• Euler's number, \( e \), as a base, provides a unique mathematical behavior perfect for continuous growth models.
• Exponential functions have the natural inverse logarithm, meaning they reverse each other: \( e^{\ln x} = x \).
• The derivative of \( e^x \) is itself, \( e^x \), which simplifies numerous calculus operations.

These properties are important because they allow exponential functions to be manipulated and solved. Their relationship to the logarithmic function provides the basis for solving complex equations involving growth or decay.

Inverse functions

Inverse functions are a crucial concept in algebra and calculus. Inverse functions essentially "undo" each other: applying one function followed by its inverse returns the original value. This is mathematically represented as \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

For exponential functions and natural logarithms, this relationship is expressed as \( e^{\ln x} = x \) and \( \ln(e^x) = x \). These identities are incredibly useful in solving equations where a variable is in a logarithmic or exponential form, allowing simplification and isolation of the variable.

Using the properties of inverse functions, you can seamlessly transition between exponential and logarithmic forms, which is indispensable for solving and understanding many types of equations in mathematics.