Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Use the definition of a logarithm to write the equation in logarithmic form. For example, the logarithmic form of \(2^{3}=8\) is \(\log _{2} 8=3$$e^{-x}=2\)

Step-by-Step Solution

Answer

The logarithmic form of the given equation \(e^{-x}=2\) is \(\ln 2 = -x\).

Step 1: Base Identification

Identify the base of the exponential expression. In the equation \(e^{-x}=2\), the base of the exponent is \(e\).

Step 2: Exponent Identification

Identify the exponent in the equation. In the equation \(e^{-x}=2\), the exponent is \(-x\).

Step 3: Result Identification

Identify the result or the number you get after the exponentiation. In the equation \(e^{-x}=2\), the result is \(2\).

Step 4: Convert to Logarithmic Form

Convert to logarithmic form using the rule log_base(result) = exponent. The given equation transforms in logarithmic form as: \(\ln 2 = -x\). Please note \(\ln\) represents natural logarithm which has base \(e\).

Key Concepts

Definition of Logarithm

When studying mathematics, students often encounter the concept of logarithms, which can be perplexing at first. However, understanding the definition of a logarithm is crucial to grasping more complex algebraic operations. Put simply, a logarithm answers the question: 'To what exponent must we raise a specific base to obtain a certain number?'

To elaborate, if we have an equation in exponential form such as, \( b^x = y \), where \( b \) is the base, \( x \) is the exponent, and \( y \) is the result, the equivalent logarithmic form would be \( \text{log}_b(y) = x \). This denotes that the logarithm of \( y \) with base \( b \) equals \( x \). The logarithm serves as a bridge between the exponential form and a form that is often easier to work with, especially when dealing with multiplication and division of exponential numbers.

Exponential Expressions

The power of exponentiation is a fundamental building block in the world of mathematics. Exponential expressions are written as \( b^x \), where \( b \) is the base and \( x \) is the exponent. This notation means that the base \( b \) is multiplied by itself \( x \) times.

Taking a specific example, if we have \( 3^4 \), this means that \( 3 \) is being multiplied by itself \( 4 \) times (3 x 3 x 3 x 3), resulting in \( 81 \). The step-by-step solution provided shows us that in converting an exponential expression to logarithmic form, we first identify the base \( e \), then the exponent \( -x \), and finally the result \( 2 \). This process enables us to rewrite the expression in a way that reveals the exponent on its own, making certain calculations easier and more intuitive.

Natural Logarithm

Among all the different types of logarithms, the natural logarithm often appears in higher-level mathematics and across various scientific disciplines. We denote the natural logarithm as \( \text{ln}(x) \) and its base is the irrational number \( e \) which is approximately equal to 2.71828. This special number \( e \) has profound implications in continuous growth models, physics, and even financial calculations.

Natural logarithms are particularly useful when dealing with growth processes such as population growth, compound interest, or radioactive decay. Our example \( e^{-x} = 2 \) uses the natural logarithm to restate the relationship between \( e \) and its exponent. The logarithmic form \( \text{ln}(2) = -x \) provides clarity: to find the exponent that describes the growth process, we calculate the natural logarithm of the number we're given. The beauty of the natural logarithm lies in its simplicity and the ease with which it can be manipulated in equations dealing with continuous growth.