Problem 40 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. \(1-\frac{1}{3} e^{5 x}=-5\)

Step-by-Step Solution

Answer

The solution to the equation is \( x \approx 0.5781 \).

Step 1: Isolate the Exponential Term

Start by isolating the exponential term on one side of the equation. The equation is \(1-\frac{1}{3}e^{5x}=-5\). Begin by subtracting 1 from both sides: \(-\frac{1}{3}e^{5x} = -6\).

Step 2: Eliminate the Coefficient

To eliminate the coefficient of \(-\frac{1}{3}\), multiply both sides by \(-3\): \( e^{5x} = 18\).

Step 3: Solve for the Exponent

To solve for \(x\), take the natural logarithm (\(\ln\)) of both sides: \( \ln(e^{5x}) = \ln(18)\). This simplifies to \(5x = \ln(18)\).

Step 4: Solve for x

Divide both sides by 5 to solve for \(x\): \( x = \frac{\ln(18)}{5}\).

Step 5: Calculate and Round the Value

Compute \(\ln(18)\) using a calculator, which is approximately 2.8904. Therefore, \( x = \frac{2.8904}{5} = 0.5781\). Round 0.5781 to the nearest ten-thousandth, which gives 0.5781.

Key Concepts

Exponential Function

An exponential function is a type of mathematical function in which a constant base is raised to a variable exponent, such as in the function \(f(x) = a^x\). In the given exercise, \(e^{5x}\) serves as the exponential part, where \(e\) is the base. Here, \(e\) is a special number known as Euler's number, approximately equal to 2.71828. This number arises naturally in problems of growth and decay, making it especially important in calculus and natural sciences.
In this exercise, the term \(e^{5x}\) reflects a transformation of the variable \(x\). As \(x\) changes, \(e^{5x}\) grows or shrinks exponentially. Exponential functions have unique properties, for instance, they never result in negative function values and rapidly increase as the exponent becomes larger.
Key features of exponential functions include:

• Continuous and smooth curves without abrupt changes.
• A horizontal asymptote, which is a line that the graph approaches but never touches.
• The rate of change increases or decreases rapidly.

Understanding these properties helps in grasping how changes in \(x\) lead to proportional changes in \(e^{5x}\), essential for isolating and solving for \(x\) in exponential equations.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the inverse operation of the exponential function with base \(e\). It answers the question: 'To what power must \(e\) be raised to produce a given number?' This makes it very useful when solving equations involving \(e\), such as the one in our exercise.
In step-by-step solutions, applying the natural logarithm allows us to shift the exponential term into a linear equation. For instance, from \(e^{5x} = 18\), applying \(\ln\) to both sides results in \(\ln(e^{5x}) = \ln(18)\).
When the natural logarithm is applied, it simplifies the exponent, utilizing the property \(\ln(a^b) = b \cdot \ln(a)\). Hence, \(\ln(e^{5x})\) simplifies directly to \(5x\) because \(\ln(e) = 1\).
Key aspects of the natural logarithm function include:

• It is undefined for non-positive values.
• Provides a convenient tool to linearize exponential growth models.
• The graph of \(y=\ln(x)\) is a curve increasing as \(x\) increases, with a vertical asymptote at \(x=0\).

Mastering natural logarithms is crucial for managing and manipulating exponential functions in algebra and calculus.

Isolation of Variables

Isolation of variables is a fundamental algebraic technique used to solve equations. The main objective is to rearrange the equation to get the unknown variable \(x\) by itself on one side of the equation. By doing this, you transform a complex equation into a simple form, making it possible to solve for \(x\).
In our exercise, isolating \(e^{5x}\) was the crucial first step. Starting with the equation \(1-\frac{1}{3} e^{5x}=-5\), we systematically rearrange it:

• First, subtract 1 from both sides to simplify the left-hand side: \(-\frac{1}{3}e^{5x} = -6\).
• Next, eliminate the coefficient by multiplying both sides by -3, resulting in \(e^{5x} = 18\).

Each rearrangement simplifies the equation, peeling away layers until \(e^{5x}\) is isolated. This simplified view makes it possible to apply logarithmic methods, thus solving for \(x\).
This process of isolation is a methodical and logical step-by-step approach used broadly in algebra to tackle not just exponential equations but any equation requiring variable solutions. Mastering it helps greatly in understanding and solving mathematical problems effectively.