Problem 24 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$250-(1.87)^{x}=0$$

Step-by-Step Solution

Answer

The algebraic solution to the exponential equation \(250 = (1.87)^x\) is \(x \approx 21.84\), which can be confirmed using a graphing calculator. To solve algebraically, we took the natural logarithm of both sides, simplified using logarithmic properties, and isolated the variable x. The graphing calculator shows the intersection point of the line \(y = 0\) and the function \(y = 250 - (1.87)^x\) at an x-coordinate of approximately 21.84, confirming our algebraic solution.

Step 1: Add \((1.87)^x\) to both sides of the equation

Add \((1.87)^x\) on both sides of the given equation to isolate the term on the left side: $$250 - (1.87)^x + (1.87)^x = (1.87)^x$$ Now, the equation becomes: $$250 = (1.87)^x$$

Step 2: Take the natural logarithm (ln) of both sides

To make the variable \(x\) easy to isolate, we take the natural logarithm (ln) of both sides of the equation: $$\ln(250) = \ln((1.87)^x)$$

Step 3: Use the property of logarithms to simplify the equation

Using the property of logarithms, we can rewrite the right side of the equation as: $$\ln(250) = x\ln(1.87)$$

Step 4: Solve for \(x\)

Now, we want to isolate \(x\). To do that, divide both sides of the equation by \(\ln(1.87)\): $$x = \frac{\ln(250)}{\ln(1.87)}$$ Now, we can use a calculator to find the approximate value of \(x\): $$x \approx 21.84$$ The algebraic solution is \(x \approx 21.84\) which can be checked using a graphing calculator.

Step 5: Graphing Calculator Check

To check our solution using a graphing calculator, we need to graph the given function: $$y = 250 - (1.87)^x$$ Now, use the graphing calculator to find the \(x\)-value of the intersection point between the line \(y = 0\) and our function. The point should have an \(x\)-coordinate of approximately 21.84. This confirms our algebraic solution.

Key Concepts

Algebraic Solutions

When faced with an exponential equation like \(250 - (1.87)^x = 0\), solving it algebraically might seem tricky at first. But by following a systematic approach, you can easily manage it.

First, rearrange the equation to isolate the exponential term. In this case, add \((1.87)^x\) to both sides, yielding \(250 = (1.87)^x\). This sets you up to work directly with the exponential term.

Next, you'll need a way to solve for the exponent \(x\). That's where logarithmic functions come into play, as they allow you to manipulate exponential equations into a solvable form.

By applying the natural logarithm, \(\ln\), on both sides, the equation becomes \(\ln(250) = \ln((1.87)^x)\). Through logarithmic properties, simplify the right side to \(\ln(250) = x\ln(1.87)\).

Finally, isolate \(x\) by dividing both sides by \(\ln(1.87)\), giving \(x = \frac{\ln(250)}{\ln(1.87)}\). Using a calculator, this arrives at \(x \approx 21.84\). This process showcases how you can transform and solve exponential equations using algebraic methods.

Logarithmic Functions

Logarithmic functions play a crucial role in solving exponential equations, especially when exponents are involved.

• Definition: A logarithm answers the question: to what exponent must we raise a base to obtain a certain number?
• Natural Logarithms: These are logarithms to the base \(e\), a mathematical constant approximately equal to 2.718. Represented as \(\ln\).
• Properties: One essential property is \(\ln(a^b) = b\ln(a)\). This property allows us to bring down exponents, making them more manageable.

When you have an equation like \(250 = (1.87)^x\), taking the natural logarithm of both sides transforms it into a linear form \(\ln(250) = x\ln(1.87)\).

This manipulation is useful because solving linear equations is generally straightforward. It effectively turns a complex problem into one that can be easily handled.

Graphing Calculators

Graphing calculators are powerful tools for checking algebraic solutions, especially with exponential functions.

By graphing the original function, \(y = 250 - (1.87)^x\), and the line \(y = 0\), you can visually confirm the solution.

• Graphing the Function: Enter the equation in your calculator to see its graph. This helps you understand the behavior and intersections of the function.
• Finding Intersections: The point where the graph touches \(y = 0\) corresponds to the solution of the original equation.
• Verification: For our problem, you should see that the intersection point is at \(x \approx 21.84\), verifying the algebraic answer.

By leveraging technology, graphing calculators make it easier to visualize and confirm solutions, providing a reliable second method to ensure accuracy.