Problem 8 - Calculus of a Single Variable

Question

Solving an Exponential or Logarithmic Equation In Exercises 1-16, solve for \(x\) accurate to three decimal places. $$ 100 e^{-2 x}=35 $$

Step-by-Step Solution

Answer

The value of \(x\) accurate to three decimal places is approximately 0.566

Step 1: Divide Both Sides by 100

The first step in solving this equation is to simplify it. Divide both sides of the equation by 100 to isolate \(e^{-2x}\) on one side of the equation.This results in \(e^{-2 x} = 0.35\)

Step 2: Take the Natural Logarithm of Both Sides

To get \(x\) out of the exponent, take the natural logarithm of both sides, using the property that \(\ln (e^{a}) = a\). This results in \(-2x = \ln(0.35)\)

Step 3: Final Step to Isolate \(x\)

Finally, divide both sides by -2 to solve for \(x\). This gives \(x = \frac{-\ln(0.35)}{2}\)

Key Concepts

Isolate Exponential Variables

To effectively solve an equation containing an exponential term, the first aim is to isolate the variable involved in the exponent. This can be approached by performing basic algebraic operations such as addition, subtraction, multiplication, or division to both sides of the equation. When dealing with exponents, these operations help us to simplify the equation so that the exponential term stands alone on one side. For example, in the given exercise, dividing by 100 simplified the equation to \(e^{-2x} = 0.35\), providing a clear view of the exponential variable. This step is critical because it lays the groundwork for applying logarithmic functions to both sides of the equation, which is the next strategic move in solving for the variable.
Remember that balancing the equation by performing the same operation on both sides maintains the equation's equality, ensuring an accurate solution. Isolating the exponential variable is a key step in not only simplifying the equation but also setting up for the application of logarithms, which is crucial in the process of solving exponential equations.

Properties of Logarithms

Understanding the properties of logarithms is indispensable when solving exponential equations. Logarithms are the inverses of exponentials, making them powerful tools for dealing with exponential variables. A core property used in the solution process is \(\text{log}_b(b^x) = x \), which states that the logarithm of a number with its base results in the exponent itself. The natural logarithm, denoted as \( \text{ln}(x) \), is a special case where the base is \(e\), Euler's number. In our example, taking the natural logarithm of both sides — \( \text{ln}(e^{-2x}) = \text{ln}(0.35) \) — allows us to apply the property \(\text{ln}(e^a) = a\) to obtain \(-2x = \text{ln}(0.35)\).
This property reflects how logarithms can simplify complex exponential expressions and make it possible to solve for the variable locked within the exponent. The student's ability to recognize when and how to apply these properties directly impacts their success in solving exponential logarithmic equations.

Natural Logarithm Applications

The natural logarithm, commonly abbreviated as \( \text{ln} \), is one of the most important concepts in mathematics, particularly when dealing with natural growth processes or time decay in physics and finance. Its base is the irrational number \(e\), approximately equal to 2.71828, which arises naturally in numerous mathematical contexts. In our exercise, the application of the natural logarithm to both sides of the equation was the pivotal step that made it possible to de-exponentiate the variable \(x\). After applying the natural logarithm, we arrive at a linear equation that can be solved using simple algebra. The natural logarithm is also frequently used in calculus for differentiation and integration involving exponential functions, as well as in real-life scenarios like computing compound interest, analyzing growth models, and interpreting the rate of decay in radioactive substances.
By grasping the concept of the natural logarithm and its practical applications, students not only advance their problem-solving skills but also appreciate its widespread relevance in various scientific and financial fields.