Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$7-2 e^{x}=1$$

Step-by-Step Solution

Answer

The solution to this equation is \(x = 1.099\)

Step 1: Simplify the equation

Start by making the equation simpler. As per the given equation \(7-2 e^{x} =1\), add \(2e^x\) on both sides and subtract 1 from both sides, which gives us: \(2e^{x} = 7 - 1\), which simplifies to \(2e^{x} = 6\).

Step 2: Isolate the exponential term

Now, we want to isolate \(e^x\). So, divide both sides of the equation by 2. This gives us: \(e^{x} = 6 / 2\), which simplifies to \(e^{x} = 3\).

Step 3: Solving for x using logarithm

Since we've isolated \(e^x\), let's now solve for \(x\) using natural logarithm (because the base of exponent is e). Applying natural logarithm to both sides gives us: \(\ln(e^{x}) = \ln(3)\). Since the natural log and the exponent 'cancel out' (as they are inverse operations), this leaves us with \(x = \ln(3)\).

Step 4: Calculating the value of x

Since we've isolated \(x\), let's plug \(\ln(3)\) into a calculator and round to three decimal places. This gives us: \(x = 1.099\).

Step 5: Verify the solution using a graphing utility

To verify the solution graphically, plot the equation \(y=7-2e^{x}\) on a graphing utility and note where the graph intersects with the line \(y=1\). You will observe that this occurs close to \(x = 1.099\), which we calculated algebraically.

Key Concepts

Solving Exponential Equations Algebraically

When dealing with an exponential equation, the goal is to find the value of the variable that makes the equation true. Algebraic manipulation is key to solving these types of equations. The first step to solve an exponential equation like 7 - 2e^{x} = 1 is to simplify the equation. Simplification involves rearranging the terms to isolate the exponential part of the equation.

To do this effectively, you'll typically perform operations such as adding or subtracting terms on both sides of the equation, as well as multiplying or dividing by numbers that will help you to isolate the exponential term. As we observe, by adding 2e^{x} to both sides and subtracting 1 from 7, we simplify our problem to 2e^{x} = 6, which is a cleaner look at our exponential hurdle.

Isolate the Exponential Term

Once we've simplified our equation, we come to the crucial step of isolating the exponential term. This makes it easier to solve for the variable. In our example, we are left with 2e^{x} = 6 after simplification. To isolate e^{x}, division is our tool of choice. We divide each side of the equation by 2, which yields e^{x} = 3.
x, from the confines of its exponential fortress.

Natural Logarithm Application

Finally, we utilize the natural logarithm to solve for x. This step leverages one of the most important relationships in mathematics: the natural logarithm is the inverse function of the exponential function with base e. Applying the natural logarithm on both sides of e^{x} = 3, we write \(\ln(e^{x}) = \ln(3)\).

Thanks to the inverse property, the complex equation simplifies to x = \ln(3). Calculating this value gives us an approximate result of x = 1.099 after rounding to three decimal places. This demonstrates how algebraic manipulation, coupled with the properties of logarithms, provides an elegant avenue to solve what could initially seem like a daunting exponential problem.