Question

Solve for \(x\). $$ e^{-2 x}=e^{5} $$

Short answer

The solution to the equation is \(x=-2.5\).

Step-by-step solution

Identify the properties of exponential functions

An important property of exponential functions is that, if the bases are equal, the exponents must be equal. So if \(e^{A} = e^{B}\), then \(A = B\). This is because an exponential function is one-to-one, which means it never takes the same value at two different points.

Apply the property to the given equation

We have the equation \(e^{-2x}=e^{5}\). Here the base of both sides is \(e\). So we can equate the powers of \(e\). Meaning, \(-2x = 5\).

Solve for \(x\)

When we have \(-2x = 5\), to solve for \(x\), we divide each side by \(-2\). So \(x = 5 / -2 = -2.5\).

Key concepts

Properties of Exponential Functions

Exponential functions are fundamental in mathematics, especially when modeling growth and decay processes. One distinct property of exponential functions is that they are one-to-one, meaning each input (or x-value) corresponds to exactly one output (or y-value). Because of this property, if we encounter two exponential expressions with the same base and they are equal, then their exponents must also be equal. This principle is essential when solving exponential equations like the given exercise.

More properties include the fact that the graph of an exponential function never touches the x-axis; it asymptotically approaches the axis as x becomes more negative. Additionally, any exponential function with a base greater than 1 will always increase, and if the base is between 0 and 1, it will decrease. These properties help us understand the behavior of exponential functions and provide the foundation for solving equations involving them.

Exponential Function Graph

Visualizing exponential functions can be crucial for understanding their behavior. The graph of an exponential function like \(y = e^x\) is a smooth curve that gets steeper as x increases if the base is greater than 1. For negative x-values, the graph approaches the x-axis but never crosses it, illustrating the concept of an asymptote—an invisible boundary the function will never reach.

In the context of our equation, \(e^{-2x} = e^5\), we can think of the graph of \(y = e^{-2x}\) as a reflection of \(y = e^{2x}\) in the y-axis, and then stretched vertically. Understanding this graphical behavior helps clarify why, when solving these equations, we look for points where the graphs of two exponential functions intersect, which corresponds to their exponents being equal when the bases are the same.

Calculating with Exponents

When calculating with exponents, it's important to understand the rules that govern their manipulation. For example, when multiplying terms with the same base, we add the exponents. When dividing, we subtract the exponents. An exponent of 0 means the term equals 1, regardless of the base (except when the base is also 0), and a negative exponent indicates the reciprocal of the base raised to the positive exponent.

In our exercise, we use these rules when equating the exponents of expressions with the same base, \(e\). We know that \(e^0 = 1\), \(e^{-2x}\) implies \(1/e^{2x}\), and if any term with an exponent equals another, their exponents should be set equal to solve for the variable. This is how we deduce that \(-2x = 5\) from the original equation \(e^{-2x} = e^5\) and subsequently solve for the value of \(x\). Understanding these principles allows for efficient and accurate calculation when working with exponents.