Problem 38 - Precalculus

Question

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$\frac{4}{3-e^{3 x}}=8$$

Step-by-Step Solution

Answer

The solution is approximately \(x = 0.305\).

Step 1: Remove the Fraction

Start by multiplying both sides of the equation by the denominator \(3 - e^{3x}\) to eliminate the fraction. This gives:\[4 = 8(3 - e^{3x})\]

Step 2: Distribute and Simplify

Distribute the 8 on the right-hand side:\[4 = 24 - 8e^{3x}\]Next, subtract 24 from both sides to isolate the term with the exponent:\[-20 = -8e^{3x}\]

Step 3: Isolate the Exponential Term

Divide both sides by -8 to isolate the exponential term:\[e^{3x} = rac{20}{8} = 2.5\]

Step 4: Solve for x Using the Natural Logarithm

Take the natural logarithm of both sides to solve for \(x\):\[3x = ext{ln}(2.5)\]Then, divide by 3 to solve for \(x\):\[x = rac{ ext{ln}(2.5)}{3}\]

Step 5: Calculate the Solution

Use a calculator to find the value:\[x \approx rac{ ext{ln}(2.5)}{3} \approx 0.305\]Thus, \(x\) rounded to three decimal places is 0.305.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is the inverse operation of exponentiation with base \(e\), where \(e\) is approximately 2.718. It's a fundamental concept in solving exponential equations, especially when the unknown variable appears in the exponent.

• The natural logarithm helps to 'undo' exponentiation: if you know \(e^x\), you can find \(x\) by calculating the natural logarithm of both sides, i.e., \(\ln(e^x) = x\).
• It's important for converting exponential equations into linear ones that can be solved using basic algebra.

For example, if you have an equation like \(e^{3x} = 2.5\), take the natural logarithm to get \(3x = \ln(2.5)\). This allows for simpler manipulation and solving for \(x\) later on.

Solving Equations

Solving equations is a method needed to find the value of a variable that makes an equation true. When dealing with exponential equations, the process often involves isolating the variable found within an exponent.

• Start by simplifying the equation: eliminate fractions or distribute terms as required.
• Then, isolate the exponential expression if it's not already by itself.
• Finally, apply mathematical operations like taking logarithms to simplify further.

For example, begin by distributing and simplifying, then isolate the exponential term, and finally use logarithms to find the solution. This systematic approach ensures accurate results.

Fraction Elimination

Fraction elimination simplifies equations by removing fractions completely, making the equation easier to handle. In solving exponential equations, eliminating fractions is often the first crucial step.

• This can be done by multiplying every term by the denominator of the fraction, effectively clearing it from the equation.
• Once the fraction is eliminated, the equation may then be simplified further by combining like terms or distributing constants.

In the given exercise, the expression \(\frac{4}{3-e^{3x}}=8\) required multiplying both sides by \(3-e^{3x}\) to remove the fraction, leading to a more straightforward equation to solve.

Isolating Exponents

Isolating exponents is crucial for solving exponential equations. The goal is to get the exponential term by itself on one side of the equation.

• This might involve operations like dividing terms or moving other numbers across the equality sign.
• Once isolated, the exponential term can be addressed directly, often using logarithms to simplify and solve.

In the problem at hand, after eliminating fractions and simplifying, the equation becomes \(e^{3x} = 2.5\). Here, the exponential term \(e^{3x}\) is isolated, setting the stage for using logarithms to solve for \(x\). It’s a step-by-step process requiring careful manipulation of terms to ensure the correct answer.