Question

In Exercises, solve the equation for \(x\). $$ e^{x}=1 $$

Short answer

The solution to the equation \(e^{x} = 1\) is \(x = 0\).

Step-by-step solution

Analyze the equation

The given equation is \(e^{x} = 1\). This is an exponential equation as it involves the constant \(e\) raised to a power \(x\). It's important to note that when \(x = 0\), \(e^{x} = 1\). That is because any number to the power of 0 is 1.

Solve for x

Comparing the given equation \(e^{x} = 1\) with the result from Step 1, we find that \(x\) must equal 0. Thus, \(x = 0\) is the solution to the given equation.

Key concepts

Exponential Equation

An exponential equation involves an unknown variable in the exponent, typically written in the form of \(a^x = b\), where \(a\) is a nonzero base, \(x\) is the exponent, and \(b\) is the constant. Solving such equations usually revolves around manipulating the equation using the properties of exponents until you can isolate the variable \(x\).

When solving the exercise \(e^{x}=1\), we recognize it as an exponential equation with \(e\) as the base, \(x\) as the exponent, and \(1\) as the constant. This particular equation is a foundational example because it involves the natural exponential function, which has a base of the mathematical constant \(e\). Remember, any nonzero number raised to the power of zero equals one, so when faced with the equation \(e^{x} = 1\), we deduce that \(x\) must be zero to satisfy the equation. This logical step is crucial for students' understanding of how to solve such problems.

Natural Exponential Function

The natural exponential function is a specific type of exponential function where the base is the mathematical constant \(e\), an irrational number approximately equal to 2.71828. It is denoted as \(e^x\) and has unique properties that make it extremely important in mathematics, especially in areas such as calculus, complex analysis, and differential equations.

One distinctive feature of the natural exponential function is that its rate of growth is proportional to its value. That means, the function \(e^x\) grows faster as \(x\) increases and conversely, it approaches zero but never quite reaches it, as \(x\) becomes more negative. The exercise provided (\

Properties of Exponents

Understanding the properties of exponents is essential for solving exponential equations effectively. These properties or 'rules' help simplify and solve equations involving exponents without much compulsion. Here are the key properties:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\) says that when multiplying two of the same bases, you can add their exponents.
• Power of a Power: \( (a^m)^n = a^{mn}\) indicates that when raising a power to another power, multiply the exponents.
• Power of a Product: \( (ab)^n = a^n b^n\) shows that you can distribute the exponent to all factors inside a product.
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n}\) tells us that when dividing two of the same bases, you can subtract the exponents.
• Zero Exponent: \(a^0 = 1\), regardless of the value of \(a\), as long as \(a \eq 0\). This property is directly utilized in solving the given textbook exercise \(e^x = 1\).
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\) implies that a negative exponent indicates the reciprocal of the base raised to the corresponding positive power.

Applying these exponent properties allows us to untangle complex exponential expressions and reach the solution with clarity. In our example, knowing that any base, including \(e\), raised to the power of zero gives the result of one, we can swiftly conclude that \(x = 0\) is the solution to \(e^x = 1\).