Problem 80 - Precalculus : Building Concepts and Connections

Question

Solve each equation graphically and express the solution as an appropriate logarithm to four decimal places. If a solution does not exist, explain why. $$e^{t}=6$$

Step-by-Step Solution

Answer

The solution to the equation after rounding to four decimal places is \(t = 1.7918\).

Step 1: Initial Visualization

We can start by plotting the two functions \(y=e^t\) and \(y=6\).

Step 2: Finding Intersection Point

Look for the point where these two graphs intersect. This intersection point corresponds to the solution of the equation.

Step 3: Using Algebraic Solution for Confirmation

In order to confirm the graphical solution algebraically, we take the natural logarithm on both sides of the given equation. Hence, we have: \(ln(e^{t}) = ln(6)\) and since \(ln(e^{t})\) simplifies to \(t\), we find \(t = ln(6)\).

Step 4: Rounding to Four Decimal Places

Calculate the numerical value of \(ln(6)\) and round it to four decimal places.

Key Concepts

Graphical Solutions

Solving equations graphically involves looking at the visual representation of functions on a coordinate plane. When graphing, each function is illustrated as a distinct curve or line, and you look for where they meet.
In this example, we need to solve the equation \(e^t = 6\) graphically.

• First, plot the curve for \(y = e^t\). This is an exponential curve that grows rapidly as \(t\) increases.


• Next, plot the horizontal line representing \(y = 6\). It's a constant value, meaning it remains flat without any slope.

The core idea of the graphical method is identifying the point where these two graphs intersect, as this gives us a visual clue about the solution of the equation. This intersection is then used to extract the numerical solution, often serving as a valuable step before formal verification through algebraic or numeric means.It is especially helpful when the solution isn't readily apparent from the equation alone.

Intersection Points

Intersection points are crucial in solving equations graphically. These are where the graphs of two functions meet on a coordinate plane. For the equation \(e^t = 6\), the intersection occurs when the values of the functions are equal.
The intersection point here shows a value along the \(t\)-axis that satisfies both \(y = e^t\) and \(y = 6\) simultaneously.To determine the intersection point:

• Look for the specific value of \(t\) where the function \(y = e^t\) equals the constant function \(y = 6\).


• In a graph, this occurs where the exponential curve \(e^t\) intersects the horizontal line \(y = 6\).

The coordinates of this point not only give a graphical solution but directly relate to the algebraic solution. Once identified correctly, use it to verify the solution through another method, such as algebra, to ensure accuracy.

Natural Logarithm

The natural logarithm, denoted as \(ln\), is a cornerstone in solving equations involving exponential functions. It has a unique base, \(e\), approximately equal to 2.71828. When we need to solve an equation like \(e^t = 6\) algebraically:

• The goal is to isolate \(t\). Applying the natural logarithm to both sides allows us to do this effectively.


• The property of logarithms, \(ln(e^t) = t\), simplifies the problem significantly. This property stems from the fact that the natural logarithm and the exponential function are inverse operations.
• Taking \(ln(6)\) yields a numerical value that corresponds to the value of \(t\). In this case, \(t = ln(6)\).

Finally, to meet any precision requirements, calculate the value of \(ln(6)\) and round it to four decimal places. This accuracy is crucial for precise applications in mathematics and sciences.Using the natural logarithm in such operations is a strategic move due to its simplicity and effectiveness in handling equations involving exponentials.