Problem 30 - Precalculus with Limits

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(1000 e^{-4 x}=75\)

Step-by-Step Solution

Answer

After crunching the numbers, you should find that \(x \approx 0.868\) when rounded to three decimal places.

Step 1: Rewrite the equation

Start solving the given equation by dividing both sides by 1000 to get \(e^{-4 x} = 0.075\)

Step 2: Apply Natural Logarithm

Next step, apply the natural logarithm to both sides to get rid of the exponential function on the left side. We will have \(-4 x = ln(0.075)\). Remember that \(ln(e^a) = a.\)

Step 3: Solve For x

Finally, solve for the variable x by dividing both sides by -4: \(x = -ln(0.075) / 4\). This can be further calculated to a numerical value when desired.

Key Concepts

Natural Logarithm

The natural logarithm is a mathematical function that is the inverse of the exponential function when the base is the natural number e, approximately equal to 2.71828. Often denoted as \(ln\)), the natural logarithm of a positive real number x is defined as the area under the curve y=1/t from 1 to x. One of the most important properties of natural logarithms is that the natural logarithm of the exponential function e to the power of a number, that is \(ln(e^{a})\)), is equal to that number a. This unique feature is particularly useful in solving exponential equations algebraically, as seen in the original exercise.

Understanding \(ln\)) is critical when dealing with exponential decay or growth problems in various fields such as science, finance, and statistics. In our context, taking the natural logarithm of both sides of an exponential equation allows us to extract the exponent as a factor, a technique that simplifies the solution process, as it isolates the variable of interest.

Exponential Function

The exponential function is a mathematical expression where a constant base, usually e, is raised to a variable exponent. In general form, an exponential function appears as \(y=b^{x}\)), where the base b is a positive real number. When b is the natural number e, the function describes natural exponential growth or decay, depending on the sign of the exponent.

The exponential function is characterized by its properties: it is always positive, it grows rapidly as x increases, and has a constant rate of growth proportional to its size, which makes it crucial to represent continuous growth. This function is used to model many real-world situations, including population dynamics, radioactive decay, and continuously compounded interest. In the exercise, \(1000e^{-4x}=75\)), we deal with an exponential decay since the exponent -4x is negative. To solve equations involving exponential functions, we typically use logarithms because they allow us to undo the exponential and solve for the exponent directly.

Algebraic Methods

Algebraic methods refer to a set of techniques used to manipulate equations and expressions in order to solve for unknown variables. In the context of exponential equations, algebraic methods can include isolating the exponential term, using logarithms to linearize the equation, and applying the inverse operations to solve for the variable of interest.

In our original exercise, the algebraic method starts with simplifying the equation by dividing both sides by 1000. This step isolates the exponential term on one side of the equation. Next, we use the natural logarithm to cancel out the exponential function, converting the equation into a form that allows linear operations. Finally, we divide by the coefficient of x to solve for the variable. Algebraic methods are the foundation for solving not just exponential equations, but a wide array of mathematical problems, as they give a structured approach to finding solutions.