Problem 5 - Precalculus

Question

Solve the exponential equations exactly for \(x\). $$e^{2 x+3}=1$$

Step-by-Step Solution

Answer

x = -\frac{3}{2}.

Step 1: Understand the Equation

First, we need to understand the given equation. We have an exponential equation in the form \(e^{2x+3} = 1\). Our goal is to isolate and solve for \(x\).

Step 2: Recognize Base e Property

The equation \(e^{2x+3} = 1\) implies that the exponent must be zero because any number raised to the power of zero is 1. Therefore, \(2x + 3 = 0\).

Step 3: Isolate the Term with x

Solve for \(x\) by isolating it in the equation \(2x + 3 = 0\). First, subtract 3 from both sides to get: \(2x = -3\).

Step 4: Solve for x

Divide both sides by 2 to solve for \(x\). This gives: \(x = \frac{-3}{2}\).

Key Concepts

solving exponential equations

Solving exponential equations involves finding the value of the variable that makes the equation true. In an exponential equation, the variable appears in the exponent. For example, in the equation \( e^{2x+3} = 1 \), \( x \) is located in the exponent. The first step to solving such equations is to simplify them, often by isolating the exponential part or utilizing properties of exponents.

To isolate the variable, you can take the logarithm of both sides, but sometimes recognizing special properties helps more directly. Here, since \( e^{0} = 1 \), the exponent must be zero because any non-zero base raised to the power of zero equals 1. Therefore, converting \( 2x + 3 = 0 \) into a straightforward algebraic equation and solving for \( x \) allows finding the precise solution.

exponential functions

Exponential functions are mathematical functions of the form \( f(x) = a^{x} \), where \( a \) is a constant and \( x \) is the variable. A common example is the natural exponential function \( e^{x} \), where \( e \) is Euler's number, approximately 2.718. These functions exhibit rapid growth or decay depending on the value of the exponent.

When dealing with exponential functions like \( e^{2x+3} \), notice the combination of constants and the variable exponent. This impacts the curve's shape and the solutions when solving equations. The initial value (or y-intercept in many contexts) and the base's growth rate create unique features, such as consistent multiplication in intervals and doubling or halving behaviors determined by the base \( e \). Understanding these can simplify handling complex equations.

properties of exponents

The properties of exponents are fundamental rules that describe how exponential expressions behave. These rules include:

• Product of Powers: \( a^{m} \cdot a^{n} = a^{m+n} \).
• Quotient of Powers: \( \frac{a^{m}}{a^{n}} = a^{m-n} \), where \( a eq 0 \).
• Power of a Power: \( (a^{m})^{n} = a^{m \cdot n} \).
• Power of a Product: \( (ab)^{n} = a^{n} \cdot b^{n} \).
• Zero Exponent Rule: \( a^{0} = 1 \), where \( a eq 0 \).

These rules help in simplifying expressions and solving equations. In the problem \( e^{2x+3} = 1 \), recognizing the zero exponent rule allowed concluding \( 2x+3 = 0 \). Understanding and applying these properties can make solving exponential equations more intuitive and straightforward.