Problem 53 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to solve each equation for \(x.\) $$5=3^{x}$$

Step-by-Step Solution

Answer

The solution to the equation \(5=3^{x}\) using a graphing utility is approximately \(x = 1.465\).

Step 1: Utilize your graphing utility

First, input the both sides of the equation separately into a graphing utility as two different functions. The functions would be \(y = 3^x\) and \(y = 5\). This will yield two distinct lines on the graph correlating to each equation.

Step 2: Locate the intersection point

After having these functions displayed on your chart, look for the intersection point of the graphs. It is the point at which both equations are equivalent, and thus represents the solution to the problem.

Step 3: Interpret the intersection point

The x-coordinate at the intersection point is the answer for \(x\) that solves the equation. Record this value.

Key Concepts

Exponential Equations

Exponential equations are a crucial part of algebra that involve variables in the exponent. They can model many real-world phenomena, such as population growth, radioactive decay, or even the cooling of an object. An exponential equation is usually in the form \(a = b^x\), where \(a\) and \(b\) are constants, and \(x\) is the variable we want to solve for.

To understand an exponential equation, you should be aware of the properties of exponential functions. For instance, as \(x\) increases, the value of \(b^x\) grows rapidly (if \(b > 1\)) or decreases quickly (if \(0 < b < 1\)). This rapid growth or decay contrasts sharply with linear or polynomial functions. Hence, they are easily distinguishable on a graph.

In the given example \(5 = 3^x\), 3 is the base, and the equation seeks the value of \(x\) that makes \(3^x\) equal to 5. Traditional methods include using logarithms to solve these equations algebraically, but graphing utilities can also offer a visual solution.

Intersection Points

Intersection points occur where two graphs meet. In the context of solving equations, finding the intersection point is synonymous with finding the solution of the equation.

When you graph an equation like \(y = 3^x\) and compare it with a horizontal line like \(y = 5\), their intersection tells us the value of \(x\) where the equation \(3^x = 5\) holds true. The intersection point has an x-coordinate that equals the solution of the equation and a y-coordinate that confirms both functions equal each other.

Graphing utilities help us easily locate these points visually. This method is particularly helpful when dealing with complex equations or systems involving more than one equation.

• Plot both equations on the same graph.
• Identify where they intersect on the graph.
• The x-coordinate at this point offers the solution.

Understanding intersection points gives a clearer picture of how two functions relate, letting you interpret equations visually rather than relying solely on calculation-intensive methods.

Graphical Solutions

Graphical solutions are a dynamic approach to solving equations that rely on graphing each part of the equation. This involves using tools or calculators to plot functions, which is particularly useful for visual learners or complicated functions.

To find a graphical solution for \(5 = 3^x\), you first plot the function \(y = 3^x\) and the line \(y = 5\) on the same axis. The point at which these graphs intersect represents the solution to the equation.

• Utilize graphing calculators or software to plot the functions.
• Zoom in to accurately pinpoint the intersection.
• Read off the x-value at this intersection for your solution.

Graphical solutions offer a broader sense of the behavior of each function, providing insights that numeric solutions alone may not offer. This method can reveal multiple solutions at a glance for more complex systems, showing potential patterns or behaviors not easily visible through calculation alone.
Engaging with graphical solutions enhances understanding and problem-solving skills, as it combines visual analysis with mathematical principles.