Problem 317 - Precalculus

Question

Use logarithms to solve. \(e^{2 x}-e^{x}-6=0\)

Step-by-Step Solution

Answer

The solution is \(x = \ln(3)\).

Step 1: Identify the Substitution

Notice that the equation \(e^{2x} - e^x - 6 = 0\) can be simplified by substituting \(y = e^{x}\). This converts it into a quadratic equation: \(y^2 - y - 6 = 0\).

Step 2: Solve the Quadratic Equation

The quadratic equation is \(y^2 - y - 6 = 0\). To solve it, we can factor it as \((y - 3)(y + 2) = 0\). Thus, the solutions for \(y\) are \(y = 3\) and \(y = -2\).

Step 3: Back Substitution

Now that we have \(y = 3\) and \(y = -2\), we substitute back \(y = e^x\). This gives us the equations \(e^x = 3\) and \(e^x = -2\). However, \(e^x = -2\) has no real solution, because the exponential function is always positive.

Step 4: Solve for x

For the equation \(e^x = 3\), take the natural logarithm of both sides to isolate \(x\): \[x = \ln(3)\]. Thus, the solution is \(x = \ln(3)\).

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. In our exercise, the equation is given by \(e^{2x} - e^{x} - 6 = 0\). Such equations often involve exponential functions like \(e^{x}\), where \(e\) is the base of the natural logarithms, approximately equal to 2.718.To solve exponential equations, we sometimes need to convert them into a more manageable form. This can involve taking the logarithm of both sides of an equation, or using substitution to simplify the expression, which will make it easier to manipulate.Understanding the properties of the exponential function is crucial. For instance:

• Exponential functions are continuous and always positive.
• As the exponent increases, the value of the exponential function rises rapidly.
• The inverse operation to exponentiation, which we use in our steps, is taking the logarithm.

Successfully solving exponential equations requires careful manipulation and equivalence transformations, as we'll explore further in the sections on quadratic equations and substitution.

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, usually in the form \(ax^2 + bx + c = 0\). In our exercise, after using substitution, a quadratic equation forms: \(y^2 - y - 6 = 0\).Quadratic equations can be solved using various methods:

• Factoring: Finding two numbers that multiply to give \(c\) and add to give \(b\). Here, it factors to \((y - 3)(y + 2) = 0\).
• Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula is used when factoring is not straightforward.
• Completing the Square: Adjusting the equation to form a perfect square on one side.

In solving our equation, with the roots \(y = 3\) and \(y = -2\), we can determine the solution for \(y\) before back substitution.

Substitution Method

The substitution method is a useful technique for transforming complex equations into simpler ones, by replacing variables with another variable or expression. In our problem, we set \(y = e^{x}\), transforming the original equation into a quadratic form, which is easier to solve.Here's how substitution simplifies the process:

• Identifies repetitive elements within the equation (e.g., powers of \(e^x\)).
• Allows the use of algebraic techniques, like factoring, to solve more straightforward behavior equations.
• Facilitates finding polynomial roots without excessive complexity.

This step is crucial in bridging the original equation to a more immediate algebraic solution strategy. After solving the quadratic \(y^2 - y - 6 = 0\), we back substitute to find the solution to the original exponential equation.

Natural Logarithms

Natural logarithms are logarithms to the base \(e\), denoted by \(\ln\). They have crucial applications in solving equations involving exponential terms, like our final step where we solved \(e^{x} = 3\) by taking the natural log.Key Properties of Natural Logarithms:

• Inverses of exponential functions: \(\ln(e^{x}) = x\).
• The natural logarithm of the product \(ab\) is \(\ln(a) + \ln(b)\).
• \(\ln(1) = 0\) because \(e^0 = 1\).
• \(\ln(e) = 1\) because \(e^1 = e\).

By applying the natural logarithm to the equation \(e^x = 3\), we isolate \(x\) as \(x = \ln(3)\), solving the equation completely. Understanding natural logarithms is essential when dealing with exponential equations in algebra and calculus.