Problem 1 - Calculus of a Single Variable

Question

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ e^{\ln x}=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Recognize the Inverse Operations

The first step is to notice that the natural logarithm \(\ln x\) and the natural exponential function \(e^{\ln x}\) are inverse functions, meaning they cancel each other out. This gives us: \(e^{\ln x} = e^{\ln(4)}\).

Step 2: Apply the Properties of Exponents

Next, by properties of the exponential and logarithm, \(e^{(\ln x)} = x\) and likewise \(e^{(\ln 4)} = 4\). Therefore, our equation simplifies into: \(x = 4\).

Key Concepts

Inverse Functions

Inverse functions are pairs of functions that essentially "undo" each other. When one function is applied to the result of another, they cancel out and return the original value. In the given exercise, the functions involved are the natural logarithm \(\ln x\) and the exponential function \(e^x\). They are inverses because the exponential function \(e^{\ln x}\) will yield \(x\), and similarly, the logarithm of an exponent \(\ln(e^x)\) will also return \(x\). These operations reverse each other:

• The exponential function raises \e\ (approximately 2.718) to a power.
• The natural logarithm finds which power of \e\ gives a certain number.

Recognizing these inverse relationships can simplify complex expressions, as seen in the initial arithmetic simplification from \(e^{\ln x} = 4\) to \(x = 4\). This critical understanding allows you to solve seemingly complex exponential and logarithmic equations with ease.

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is a special logarithm with the base \e\. It is an essential concept because it naturally occurs in various mathematical and real-world phenomena. The number \e\ is an irrational constant approximately equal to 2.718. The natural logarithm answers the question: "To what power should \e\ be raised to produce this number?"

• In mathematical terms, if \(\ln x = y\), it implies that \(e^y = x\).
• It is often used in calculus due to its unique properties, like its derivative \(\frac{d}{dx}[\ln x] = \frac{1}{x}\).

In the exercise, we solve for \(x\) using the property that \(\ln x\) is the inverse of the exponential function. This inverse relationship directly simplifies equations like \(e^{\ln x} = 4\). The transformation leads us directly to \(x = 4\), emphasizing the utility of understanding the natural logarithm in solving exponential equations.

Properties of Exponents

Exponents are a way to express repeated multiplication, and the properties they have are crucial for simplifying calculations and solving equations. Understanding exponents is key to working efficiently with logarithmic and exponential equations. Here are some typical properties:

• \(e^{m+n} = e^m \cdot e^n\): Adding exponents multiplies the base numbers.
• \(e^{m-n} = \frac{e^m}{e^n}\): Subtracting exponents divides the base numbers.
• \((e^m)^n = e^{m \cdot n}\): An exponent raised to another exponent means multiply the exponents.

These rules simplify complex expressions and play a critical role when working with equations involving \(e\) and natural logarithms, like converting \(e^{\ln 4}\) into 4. In the given problem, we apply these properties, especially noting that \(e^{\ln x} = x\). Such identities not only solve mathematical queries but also enhance computational fluency, reducing what might seem complex into an understandable form.