Problem 121 - Elementary and Intermediate Algebra

Question

If \(e^{t+5}=k e^{t},\) find \(k\)

Step-by-Step Solution

Answer

The value of \( k \) is \( e^{5} \).

Step 1: Write the given equation

We are given the equation \( e^{t+5} = k e^{t} \). This is what we need to work with to find the value of \( k \).

Step 2: Apply the property of exponents

Recall the property of exponents \( e^{a+b} = e^{a}e^{b} \). Apply this property to the left hand side of the equation. Thus, \( e^{t+5} \) can be rewritten as \( e^{t} e^{5} \). The equation becomes:\[ e^{t} e^{5} = k e^{t} \]

Step 3: Isolate the constant k

As long as \( e^{t} eq 0 \), we can divide both sides of the equation \( e^{t} e^{5} = k e^{t} \) by \( e^{t} \) to solve for \( k \). After dividing, we get:\[ e^{5} = k \]

Step 4: Simplify the solution

The value of \( k \) is \( e^{5} \). Therefore, \( k = e^{5} \).

Key Concepts

Properties of Exponents

Exponents are a fundamental concept in algebra that describe how many times a number, known as the base, is multiplied by itself. Understanding the rules governing exponents is key to solving exponential equations. One important property is the addition of exponents when multiplying like bases. For example,

• when you have an expression like \( e^{a+b} \), it can be broken down into \( e^{a} \times e^{b} \).

This property allows us to neatly simplify complex expressions involving exponents. In the exercise, applying this rule transformed \( e^{t+5} \) into \( e^{t} \times e^{5} \). This decomposition is crucial because it exposes the structure of the equation, making it easier to manipulate and solve. Breaking down exponent expressions using these properties can efficiently streamline solving exponential expressions and equations.

Exponential Equations

Exponential equations are equations where variables appear as exponents. These types of equations can appear daunting due to the exponential growth involved, but understanding their structure can simplify solving them. They often include expressions like \( e^{x} \), where \( e \) is the base of the natural logarithm.

• Exponential equations usually require manipulation of the exponents to isolate the unknown.
• Identifying and applying properties of exponents is often a necessary first step.

In our example, the equation \( e^{t+5} = k e^{t} \) required rewriting the left-hand side using properties of exponents. Recognizing that both terms share the same base \( e \), we can navigate to an easier path for finding \( k \). By understanding how to work with exponential forms, the method to find solutions becomes clearer and more approachable.

Solving Equations

The process of solving equations involves finding the value of the unknown that makes the equation true. Whether dealing with linear, quadratic, or exponential equations, the main goal is always the same: isolate the variable. In the case of exponential equations, this often means reducing the equation to a simpler form where the variable can be isolated.

• To solve \( e^{t+5} = k e^{t} \), we need to isolate \( k \).
• This is accomplished by dividing both sides by \( e^{t} \), provided \( e^{t} eq 0 \).

This simplification reveals the value for \( k \): \( e^{5} \). In these cases, it's crucial to assume that the mathematical operations performed, like division, are valid—in this instance, ensuring that \( e^{t} \) is not zero. Mastering the art of manipulating and simplifying equations is essential for solving more complex problems in mathematics.