Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(3 e^{x}=9\)

Step-by-Step Solution

Answer

The solution of the equation \(3 e^{x} = 9\) rounded to three decimal places is \(x \approx 1.099\).

Step 1: Isolate the exponential function

To isolate the exponential function on one side of the equation, we first need to divide both sides of the equation by 3, giving us \(e^{x} = 3\).

Step 2: Apply the natural logarithm

The natural logarithm and exponential function are inverse functions, which simplifies the equation. Taking the natural log of both sides we get: \(\ln(e^{x}) = \ln(3)\). This simplifies to: \(x = \ln(3)\).

Step 3: Approximate the result

Since we need to give the result to three decimal places, we evaluate \(\ln(3)\) using a calculator and round it to the nearest thousandth obtaining: \(x \approx 1.099\).

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. The general form of an exponential function is f(x) = a^x, where a is a positive constant, often referred to as the base, and x is an exponent which can take any real value. Exponential functions have unique characteristics. They show rapid growth or decay, depend on whether the base is greater than one or between zero and one.

Consider e^x, which is an exponential function where the base e is an irrational and transcendental number approximately equal to 2.71828, known as Euler's number. It is a fundamental base in calculus and natural sciences because of its unique properties in growth processes and compound interest calculations. In the exercise 3e^x = 9, dividing by 3 to isolate e^x, we can see the exponential growth behavior concerning x.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is the number approximately equal to 2.71828. The natural log of a number y is the power to which e must be raised to obtain y. Mathematically, we write this as y = e^x is equivalent to ln(y) = x.

One of the most important properties of the natural logarithm is that it is the inverse function of the exponential function with the base e. Because of this inverse relationship, when you apply the natural logarithm to e^x, you're effectively 'undoing' the exponential, leaving you with the exponent, as shown in the exercise: ln(e^x) = x. This property is what makes it possible to solve exponential equations like the one in the exercise.

Approximation

Approximation in mathematics involves finding a value that is close enough to the correct answer, typically for the sake of easier computation or understanding. When dealing with irrational numbers or results of complicated functions, it's common to round to a certain number of decimal places.

In the context of our exercise, we can't express the natural logarithm of 3 as a simple fraction or exact decimal because it is an irrational number. Hence, we use approximation. When the problem asks for a three-decimal place approximation, we're being asked to round off the result ln(3) after calculating it with a calculator. This is an essential skill in both pure and applied mathematics, as it allows for practical use of complex, precise numbers in everyday computations and applications.