Question

Solve for \(x\) to three significant digits. $$e^{x}+e^{-x}=2\left(e^{x}-e^{-x}\right)$$

Short answer

\(x \approx 0.549\)

Step-by-step solution

- Expand and Rearrange the Equation

Begin by expanding the given equation: \(e^{x} + e^{-x} = 2e^{x} - 2e^{-x}\). Rearrange the terms to get all the terms involving \(e^x\) and \(e^{-x}\) on one side, leading to the equation \(e^{x} - 2e^{x} + e^{-x} + 2e^{-x} = 0\). Simplify to get \(-e^{x} + 3e^{-x} = 0\).

- Find Common Exponent and Factor

Recognize that \(e^{x}\) and \(e^{-x}\) are not like terms, so convert \(e^{-x}\) to \(\frac{1}{e^{x}}\) so that both terms contain \(e^{x}\) as a common factor. The equation becomes \(-e^{x} + 3\frac{1}{e^{x}} = 0\). Multiply through by \(e^{x}\) to clear the fraction: \(-e^{2x} + 3 = 0\).

- Solve the Quadratic Equation

The resulting equation, \(-e^{2x} + 3 = 0\), is a quadratic-like equation in terms of \(e^{x}\). Add \(e^{2x}\) to both sides to obtain \(e^{2x} = 3\). Taking the natural logarithm of both sides yields \(2x = \ln(3)\). Divide both sides by 2 to isolate and solve for \(x\): \(x = \frac{\ln(3)}{2}\).

- Calculate the Value of x

Using a calculator, compute the value of \(x\) to three significant digits: \(x = \frac{\ln(3)}{2} \approx 0.549\).

Key concepts

Logarithms

Logarithms are an essential concept in mathematics, especially when dealing with exponential equations. A logarithm answers the question: to what exponent do we need to raise a certain base number to obtain a given number? In simpler terms, if we have a relation like a^b = c, the logarithm of c with base a is b, written as log_a(c) = b. Therefore, logarithms are the inverse operation of exponentiation.

Many students find logarithms challenging because they're an abstract concept. To overcome this, visualization can be helpful. Imagine a logarithmic scale where each step on the scale represents a power of the base. This is often seen in scientific data representation where large ranges of data need to be represented compactly.

The most common logarithms are those with base 10, known as common logarithms, and those with base e (Euler's number), known as natural logarithms, denoted as ln. Natural logarithms are particularly useful in calculus, complex mathematics, and real-world applications like computing compound interest or understanding decay processes.

Equation Solving

Solving equations is a fundamental skill in algebra. To solve an equation, we need to find the value of the unknown variable that makes the equation true. The strategy to solve an equation often depends on the type of equation presented. A linear equation, for instance, has a straightforward solution, whereas a quadratic equation may require factoring, completing the square, or applying the quadratic formula.

Exponential equations involve variables in the exponent and usually require taking logarithms to solve. As seen in the original exercise, by taking the natural logarithm, we can effectively 'bring down' the exponent, making it easier to isolate the variable. Once the variable is isolated, we can solve for it algebraically. It's essential to follow a systematic approach: expand and simplify the equation, isolate the term containing the variable, and perform algebraic operations to solve for the variable. As always, it's crucial to check the solution by substituting it back into the original equation to verify that it holds true.

Natural Exponent e

The natural exponent e is a fundamental constant approximately equal to 2.71828, and it's the base of natural logarithms. It is an irrational number, meaning it cannot be written as a simple fraction, and its decimal representation goes on forever without repeating.

The number e arises naturally in various fields such as economics, physics, and engineering because it has the unique property of being the rate of growth shared by all continually growing processes. For instance, when we talk about continuous compounding interest in finance, e is used to calculate the final investment value.

Properties of e

• The function e^x is its own derivative and integral, which makes it extremely useful in calculus.
• Equations with e^x naturally relate to growth and decay processes, as in radioactive decay or population growth.

Understanding e is not just about memorizing its value; it's about recognizing its role as the base in natural logarithms and how it applies to real-world exponential growth and decay scenarios.