Problem 91 - Precalculus

Question

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

Step-by-Step Solution

Answer

The statement is true for all \( x > 0 \).

Step 1: Understand the Given Expression

The expression given is \( e^{\log x} \). We need to determine if this expression is equal to \( x \). This involves understanding the relationship between the exponential function and the natural logarithm.

Step 2: Recall the Exponential and Logarithm Relationship

Recall that the natural logarithm \( \log x \) is usually denoted as \( \ln x \), which is the logarithm with the base \( e \). There is a property of logarithms and exponentials that states \( e^{\ln a} = a \).

Step 3: Apply the Exponential-Logarithmic Property

Based on the property \( e^{\ln a} = a \), substitute \( a \) with \( x \). Thus, \( e^{\ln x} = x \). Since \( \log x \) is the same as \( \ln x \) under the given base \( e \), the expression \( e^{\log x} = x \) holds true.

Step 4: Conclusion of Truth Value

Given that \( e^{\log x} = x \) adheres to the property, the statement is true for all \( x > 0 \), since the domain of the logarithm function is only positive real numbers.

Key Concepts

Natural Logarithm

The natural logarithm is a logarithm with the base of the mathematical constant \( e \), which is approximately equal to 2.71828. It is widely used in mathematics because of its natural properties in calculus and exponential growth and decay problems. The natural logarithm of a number \( x \) is written as \( \ln(x) \), and it essentially asks, "To what power must \( e \) be raised to produce \( x \)?"

Let's break this down with an example. If \( \ln(7) \approx 1.9459 \), it means that raising \( e \) to the power of 1.9459 gives you approximately 7. This shows why the natural logarithm is essential in reversing exponential expressions.

Key features of natural logarithms include:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) because \( e^1 = e \).
• The function is only defined for positive real numbers.

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a^x \), where \( a \) is a positive real number not equal to 1. The most common base for exponential functions in mathematics is \( e \). The exponential function with base \( e \) is often denoted as \( e^x \) and it plays a crucial role in growth and decay models, as well as in many areas of calculus.

Let's dive into some characteristics:

• An exponential function grows rapidly as \( x \) increases if the base is greater than 1.
• If \( a = e \), the function is called the natural exponential function and is the inverse of the natural logarithm.
• It is continuously increasing and has a domain of all real numbers but a range of positive real numbers.

To visualize, consider the graph of \( e^x \). It starts very low at negative \( x \), crosses the point \((0,1)\), and then shoots up steeply as \( x \) becomes positive. This characteristic sharp increase is what makes exponentials so powerful in describing natural phenomena.

Properties of Logarithms

Logarithmic properties are fundamental in simplifying and solving equations that involve logarithms. These properties assist in making complex expressions more manageable. Understanding these rules allows us to maneuver between exponential and logarithmic forms effectively.

Here are some must-know properties:

• \( \log_{a}(xy) = \log_{a}x + \log_{a}y \): The log of a product is the sum of the logs.
• \( \log_{a}(x/y) = \log_{a}x - \log_{a}y \): The log of a quotient is the difference of the logs.
• \( \log_{a}(x^b) = b\cdot\log_{a}x \): The log of a power is the exponent times the log of the base.

These rules apply to all logarithms, including the natural logarithm used with base \( e \). For example, by applying the property \( e^{\ln a} = a \), we can simplify expressions where exponentials and logarithms coexist. It's the complete cycle that takes us from logarithmic form back to direct numbers while maintaining equality in equations.