Problem 20 - Precalculus with Limits

Question

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(4 e^{x}=91\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3.123\).

Step 1: Isolate exponential

Divide both sides of the equation \(4 e^{x}=91\) by 4, you receive \(e^{x}=\frac{91}{4} = 22.75\).

Step 2: Apply natural logarithm

Apply the natural logarithm on both sides of the equation. As the logarithm rule for \(ln(e^{a})=a\), you get \(x = ln(22.75)\).

Step 3: Calculate the variable

Calculate \(x\) by evaluating the natural logarithm of 22.75, and leave the result to three decimal places, you have \(x = 3.123\).

Key Concepts

Isolate the Exponential Term

Solving an exponential equation means finding the value of the variable that makes the equation true. The first key step is to isolate the exponential term. When faced with an equation such as \(4 e^{x} = 91\), the goal is to get \(e^{x}\) by itself on one side of the equality. This often involves basic algebraic operations like addition, subtraction, multiplication, or in this case, division.

Here's how to tackle it: divide both sides of the equation by 4, and you are left with \(e^{x} = \frac{91}{4}\), simplifying down to \(e^{x} = 22.75\). Isolating the exponential term is crucial because it sets up the equation for the next step where we will be applying a logarithm to both sides to solve for the variable.

Apply the Natural Logarithm

Once you have the exponential term by itself, like \(e^{x} = 22.75\), the next move is to 'undo' the exponent. This is where the natural logarithm (ln) comes into play.

Applying the natural logarithm to both sides of the equation utilizes the unique relationship between logarithms and exponents. The natural logarithm is the inverse operation of the exponential function with base \(e\). That means \(ln(e^{a}) = a\), because the logarithm asks the question, 'To what power must \(e\) be raised, to get \(a\)?' So, taking the natural logarithm of both sides of \(e^{x} = 22.75\) gives you \(ln(e^{x}) = ln(22.75)\), and due to the inverse relationship, simplifies down to \(x = ln(22.75)\).

Evaluate the Natural Logarithm

The final step in solving the exponential equation is to evaluate the natural logarithm. After applying the natural logarithm, you're now faced with \(x = ln(22.75)\). Evaluating the logarithm will give you the actual numerical value for \(x\).

To evaluate \(ln(22.75)\), you can use a calculator with the 'ln' function. This calculation will give you the power to which \(e\) must be raised to obtain 22.75. When you calculate this value, remember to round it to three decimal places, as the problem requests an approximation. In this case, you get \(x ≈ 3.123\), completing the solution to the original equation. Evaluating the natural logarithm solidifies your understanding of the relationship between exponents and logarithms in solving equations algebraically.