Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ e^{-\ln x}=0.2 $$

Step-by-Step Solution

Answer

The solution is \( x = 5 \).

Step 1: Understand the Equation Form

The given equation is \( e^{-\ln x} = 0.2 \). This involves the exponential function, where \( e \) is raised to the power of \(-\ln x\). Our task is to isolate \( x \).

Step 2: Apply the Exponent Rule

Recall the property of exponents that \( e^{\ln a} = a \). However, here we have \( e^{-\ln x} \), which simplifies to \( \frac{1}{e^{\ln x}} \) or \( \frac{1}{x} \) by the property of logarithms and exponents.

Step 3: Set the Equation in Simpler Form

Now our equation transforms to \( \frac{1}{x} = 0.2 \). We can rewrite 0.2 as a fraction, which is \( \frac{1}{5} \). Thus, \( \frac{1}{x} = \frac{1}{5} \).

Step 4: Solve for x

Since \( \frac{1}{x} = \frac{1}{5} \), it follows that \( x = 5 \). This step involves solving the equation by recognizing that if two fractions are equal, their denominators must be equal.

Key Concepts

Logarithmic Properties

Logarithms have their unique set of properties that make them astonishingly useful in dealing with exponential equations. One such property is that if you have a logarithm of a number, such as \( \ln a \), raised to the exponential \( e \), it simplifies directly back to \( a \). In mathematical terms, this is expressed as \( e^{\ln a} = a \).

This property is pivotal when converting between logarithmic and exponential forms. It essentially says that the natural exponential function and the natural logarithm function are inverse operations. When dealing with negative logarithms, as in our primary equation \( e^{-\ln x} \), this property allows us to take the reciprocal, simplifying it to \( \frac{1}{x} \). This step is crucial for reducing complex exponential expressions into simpler algebraic forms.

Exponential Functions

Exponential functions are essential components of higher mathematics and descriptions of real-world phenomena like population growth or radioactive decay. In its essence, an exponential function has the form \( a^x \), where \( a \) is a constant and \( x \) is an exponent.

The natural exponential function, denoted as \( e^x \), is particularly significant because it frequently appears in calculus and complex models due to its unique derivative properties. In a problem like \( e^{-\ln x} = 0.2 \), understanding that \( e \) is a constant (approximately 2.71828) allows us to work through transformations using its logarithmic natural properties. By recognizing \( e^{- ext{something}} \), you immediately start thinking about ways to simplify, often tapping into your knowledge of logarithm reciprocity.

Solving Equations

Solving equations is foundational and requires a strong understanding of various algebraic techniques. In the context of our equation \( e^{-\ln x} = 0.2 \), once it was simplified to \( \frac{1}{x} = 0.2 \), it called for straightforward algebra to find \( x \).

Recognizing equivalence between fractions is critical here. If \( \frac{1}{x} = \frac{1}{5} \), by knowing that two fractions are equal only if their numerators and denominators can be swapped or equated, we deduce \( x = 5 \). This process highlights how solving real-world equations can often reduce to recognizing simple patterns and applying fundamental arithmetic operations.