Problem 526 - Precalculus

Question

Use logarithms to find the exact solution for \(3 e^{6 n-2}+1=-60 .\) If there is no solution, write no solution.

Step-by-Step Solution

Answer

No solution.

Step 1: Isolate the exponential expression

The given equation is \(3e^{6n-2}+1=-60\). Start by isolating the exponential expression. Subtract 1 from both sides of the equation:\[ 3e^{6n-2} = -60 - 1 \]Which simplifies to:\[ 3e^{6n-2} = -61 \]

Step 2: Solve for the exponential

To isolate \(e^{6n-2}\), divide both sides by 3:\[ e^{6n-2} = \frac{-61}{3} \]This implies that we have an exponential expression set equal to a negative number, which is not possible since the exponential function \(e^x\) is always positive for all real \(x\).

Step 3: State no solution

Since the expression \(e^{6n-2} = \frac{-61}{3}\) is not possible, the original equation has no solution. Any exponential equation of the form \(e^x\) can never result in a negative value, which confirms that this equation has no solution.

Key Concepts

Exponential Equations

Exponential equations are equations where the variable appears in the exponent. These types of equations often involve expressions like \(e^x\) or \(a^x\), where \(a\) is a constant greater than zero.
Solving exponential equations typically requires one of two strategies:

• Rewriting both sides of the equation with the same base and then equating exponents.
• Using logarithms to bring the variable down from the exponent.

The particular challenge with these equations is dealing with the exponent, as it's not a straightforward process like other algebraic equations where the variable might be a simple part of a polynomial expression. In this exercise, the equation \(3e^{6n-2} + 1 = -60\) was transformed into an exponential equation \(e^{6n-2} = \frac{-61}{3}\).
The impossibility of this equation having a solution stems from the nature of the exponential function, which we'll explore further.

Isolation of Variables

Isolating the variable in exponential equations involves algebraic manipulation to have the exponential expression alone on one side of the equation. Initially, in our exercise, the goal was to isolate \(e^{6n-2}\).

• First, subtract constants or terms not involving the variable from both sides.
• Then, divide by any coefficients present in front of the exponential term.

In our example, the process started with \(3e^{6n-2} + 1 = -60\). By subtracting 1, the equation became \(3e^{6n-2} = -61\). Next, dividing both sides by 3, we achieved \(e^{6n-2} = \frac{-61}{3}\).
This approach successfully isolates the exponential, allowing us to assess whether a solution is feasible. When the exponential term cannot equate to a valid number (like a negative), the equation is deemed unsolvable.

Exponential Function Properties

Exponential functions \(e^x\) have unique properties that can influence how equations involving them are solved. Understanding these properties is crucial for working through exponential equations.
Key characteristics of exponential functions include:

• They are always positive for any real number \(x\).
• The function's graph is always above the x-axis, meaning it never crosses zero or becomes negative.
• They grow rapidly for positive exponents and approach zero for negative exponents, never reaching zero.

In the exercise we dealt with, one crucial property was that \(e^{6n-2}\) must be positive, hence why an attempt to set \(e^{6n-2} = \frac{-61}{3}\) results in no solution. This confirms the function cannot equal a negative number, validating our analysis from both a theoretical and practical standpoint.