Problem 116 - Algebra and Trigonometry

Question

Solve: \(9 e^{3 x}-4=32 .\) Find the solution set and then use a calculator to obtain a decimal approximation to two decimal places for the solution. (Section \(4.4,\) Example 3 )

Step-by-Step Solution

Answer

The solution of the equation \(9 e^{3x}-4=32\) is \(x = 0.46\) to a decimal approximation to two decimal places.

Step 1: Isolate the Exponential Term

We first need to isolate the exponential term \(9 e^{3x}\). Add 4 to both sides of the equation to get \(9 e^{3x} = 36\) and then divide everything by 9 to find \(e^{3x} = 4\).

Step 2: Apply Natural Logarithm

Use the natural logarithm function to both sides to get \(3x = \ln(4)\), which was possible due to the property \(\ln(e^{k}) = k\).

Step 3: Solve for x

To solve for \(x\), we divide both sides of the equation by 3 resulting in \(x = \ln(4) / 3\).

Step 4: Decimal Approximation

Finally, using a calculator, the decimal approximation of the solution to two decimal places is \(x = 0.46\) .

Key Concepts

Exponential Functions

When dealing with equations, it's crucial to understand the behaviors and properties of exponential functions. An exponential function can be defined by an equation of the form \( y = a e^{kx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828, \( a \) is a constant, \( k \) is the rate of growth (or decay if negative), and \( x \) represents the independent variable.

The key feature of an exponential function is that the rate of change increases or decreases proportionally to the value of the function itself. This is why these functions are often used to model growth processes such as population growth, compound interest, and certain types of decay. When faced with solving an equation, like \( 9 e^{3x}-4=32 \), our goal is to rewrite it so that we have the exponential part, \( e^{3x} \), on one side, enabling us to apply logarithmic functions for further simplification.

Understanding the steps to isolate the exponential expression is essential. This usually involves basic algebraic operations such as addition, subtraction, multiplication, or division to rearrange the equation. Once this is done, we're ready to employ logarithms to find the value of the unknown variable, in this case, \( x \).

Natural Logarithm

The natural logarithm is a logarithm with base \( e \), the Euler's number. It's frequently denoted as \( \ln(x) \) and speaks to the power to which we must raise \( e \) to obtain the number \( x \). For instance, if we take the natural logarithm of \( e^{3x} \), we get \( \ln(e^{3x}) = 3x \), since the natural logarithm and the exponential function are inverse operations.

In the context of solving exponential equations, using the natural logarithm is a primary tool. Once we isolate the term with \( e \), applying the natural logarithm to both sides of the equation effectively 'unwraps' the exponent, as seen in the equation \( e^{3x} = 4 \). The natural logarithm simplifies the equation to \( 3x = \ln(4) \), making it possible to solve for \( x \). Remember, this step is greatly dependent on understanding the intrinsic relationship between exponents and logarithms, and mastering their properties can significantly bolster a student's problem-solving skills in algebra and calculus.

Decimal Approximation

Once we’ve reached a point where the unknown is isolated, such as \( x = \ln(4) / 3 \), the next step is to get a decimal approximation of the solution. The exact form, like \( \ln(4) / 3 \), can be exact but often not very helpful when understanding the magnitude or applying it in real-world situations.

To obtain a decimal approximation, we typically use a calculator because the natural logarithm of a number is not a simple calculation. A calculator will give us a decimal number that represents the natural logarithm value to a certain degree of accuracy. We then divide this value by 3 to find the decimal approximation of \( x \).

In instructional settings, decimal approximations are useful when the objective is to get a tangible sense of the solution's size or when the answer is needed for further calculations. Being able to round to a specific decimal place is a common requirement, as precision can be crucial. In our example, rounding to two decimal places means we want the nearest hundredth, which in this case presents \( x \) as 0.46, a more user-friendly number that can be easily understood and utilized in various applications.