Problem 8 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$4^{3 x-5}=16$$

Step-by-Step Solution

Answer

The solution to the exponential equation \(4^{3x-5}=16\) is \(x=\frac{7}{3}\). Verify the solution using a graphing calculator by plotting the functions \(y_1 = 4^{3x-5}\) and \(y_2 = 16\) and checking for their intersection point at \(x=\frac{7}{3}\).

Step 1: Rewrite the right side of the equation with base 4

We have the equation; $$4^{3x-5}=16$$ Since both 4 and 16 are multiples of 2, we can rewrite 16 in base 4. We know: $$4^2 = 16$$ Now, rewrite the equation as: $$4^{3x-5}=4^2$$

Step 2: Equate the exponents

We have successfully managed to rewrite both sides of the equation with the same base, which means we can equate the exponents as follows: $$3x-5=2$$

Step 3: Solve for x

Now that we've equated the exponents, we can solve for x: Add 5 to both sides of the equation: $$3x=2+5$$ $$3x=7$$ Now, divide both sides of the equation by 3: $$x=\frac{7}{3}$$ The solution to the exponential equation \(4^{3x-5}=16\) is \(x=\frac{7}{3}\).

Step 4: Verify the solution using a graphing calculator

To confirm the solution, use a graphing calculator to plot the two functions: \(y_1 = 4^{3x-5}\) and \(y_2 = 16\). Find the x-coordinate of the point where these two graphs intersect. The x-coordinate should be equal to the value we found for x, which is \(\frac{7}{3}\). If the graphs intersect at this x-coordinate, this verifies that our solution is correct.

Key Concepts

Algebraic Solution

Solving exponential equations algebraically involves rewriting the equations so that both sides have the same base. The process is straightforward once you understand the basic principles:

• Identify the base on one side of the equation. For example, in the equation \(4^{3x-5}=16\), the base is 4 on the left side.
• Rewrite the other side of the equation using the same base. Since \(16\) is the same as \(4^2\), the equation can be written as \(4^{3x-5}=4^2\).
• Once both sides have the same base, equate the exponents. For instance, \(3x-5\) and 2 can be set equal to each other because their respective powers are equal when the bases are the same.
• Solve for the variable as with any linear equation: \(3x-5=2\). First, add 5 to both sides to get \(3x=7\). Then, divide both sides by 3 to isolate \(x\), resulting in \(x=\frac{7}{3}\).

By understanding how to manipulate exponents and rewrite equations with a common base, solving these types of equations becomes much more manageable.

Graphing Calculators

Graphing calculators can be a powerful tool in verifying the solutions of algebraic problems, especially for exponential equations. These devices allow you to graph functions and visually inspect intersections, confirming your algebraic solutions.To use a graphing calculator to check your work:

• Input the original functions into the calculator. For the problem \(4^{3x-5}=16\), input \(y_1=4^{3x-5}\) and \(y_2=16\).
• Graph both functions. The calculator will plot the equations and display them visually.
• Find the point of intersection. The x-coordinate at this intersection point should match your previously found solution \(x=\frac{7}{3}\).

Using a graphing calculator not only helps verify your solutions but also enhances your understanding by providing a visual representation of how equations behave. It is a useful step in confirming the correctness of your work and building confidence in your algebraic abilities.

Solving Equations

Solving equations is a fundamental aspect of algebra and an essential skill for tackling more complex mathematical problems. Exponential equations require special attention because they involve variables in the exponent position, which is different from polynomial or linear equations. Here are some tips for successfully solving equations:

• Identify the type of equation you're dealing with. Exponential equations have variables in the exponent; recognizing this helps you choose the right strategy.
• Rewrite terms whenever possible to simplify the problem. In exponential equations, rewriting terms with the same base often allows you to simplify and solve them.
• Balance the equation by performing the same operations on both sides. This approach ensures that the equation remains true as you work toward isolating the variable.
• Check your solutions. Whether through substitution back into the original equation or using a graphing calculator, verifying your answer is key to confirming its accuracy.

Mastering the art of solving different types of equations paves the way for success in further studies and offers a deeper appreciation of the problem-solving aspect of mathematics.