Problem 64 - 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. $$-14+3 e^{x}=11$$

Step-by-Step Solution

Answer

The rounded solution to the equation \(-14+3e^{x}=11\) is \(x = 2.121\).

Step 1: Rearrange the Equation

Start by getting \(e^x\) term on one side by adding 14 to both sides so the equation becomes: \(3e^x = 11 + 14\) which simplifies to \(3e^x = 25\).

Step 2: Solve for \(e^x\)

To solve for \(e^x\), divide both sides of the equation by 3 to get: \(e^x = 25/3 = 8.3333\).

Step 3: Solve for \(x\)

To solve for \(x\), you need to apply the natural logarithm on both sides. That gives you: \(x = \ln(8.3333)\). Use a calculator to find that \(x = 2.121\).

Step 4: Verification with a Graphing Utility

Plot the function \(-14+3e^{x}\) and the line \(y=11\) using a graphic calculator or software. The x-coordinate of the point of intersection will be the solution to the equation. It can be seen that this approximates to 2.121, confirming the correctness of the previous calculations.

Key Concepts

Exponential Function

The exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the given exercise, the exponential function is represented as 3e^x. This notation signifies that the number 'e' is raised to the power of 'x', and then the result is multiplied by 3. The number 'e' is approximately equal to 2.71828 and is known as Euler's number; it is a fundamental constant in mathematics, especially in calculus, due to its unique properties when used as the base for exponential functions.

Exponential functions grow at a rate proportional to their value, which makes them especially important for modeling growth processes such as population growth, compound interest, and certain physical phenomena. When solving equations that involve an exponential function, one strategy is to isolate the exponential term and then apply a logarithm to both sides of the equation to solve for the variable exponent. This is precisely the strategy that was used in the step-by-step solution provided.

Natural Logarithm

The natural logarithm, denoted by ln, is the inverse operation of taking an exponential function with base 'e'. For instance, if we have an equation e^x = a, then we can find 'x' by taking the natural logarithm of both sides, which would give us x = ln(a). The natural logarithm has the special property that ln(e) = 1, and it is widely used in mathematics, physics, engineering, and many other fields due to its natural occurrence in describing time growth processes and decay.

In the provided exercise, after isolating e^x, we apply the natural logarithm to find the value of 'x'. This transforms the problem from exponential form to a simpler linear form that we can solve with basic algebra. Using a calculator, as suggested in the instructions, is essential because natural logarithms, especially of non-integer values, are typically non-integers themselves and not easily computed without technological assistance.

Graphing Utility Verification

Graphing utilities, such as graphic calculators or software, are important tools that can help verify solutions to equations, especially when complex functions are involved. When you plot the original exponential equation and compare it with the constant or linear function it's set equal to, you can visually confirm the solution by finding the point where the two graphs intersect.

In the exercise, plotting -14+3e^x and the horizontal line y=11 and observing where they cross provides a visual verification of the algebraic solution. This step is particularly useful as it serves as both a check on the algebraic work and an opportunity to understand the behavior of the exponential function graphically. Through the usage of graphing utilities, students can not only confirm their solutions but also gain insight into the relationship between algebraic equations and their graphical representations.