Problem 13 - Beginning and Intermediate Algebra

Question

Solve the exponential equation algebraically. Then check using a graphing calculator. $$10^{-x}=5^{2 x}$$

Step-by-Step Solution

Answer

In summary, to solve the exponential equation algebraically, we followed these steps: 1. Rewrote the equation to a common base: \(2^{-x\log_{2}{10}} = 2^{2x\log_{2}{5}}\) 2. Equated the exponents and solved for x: \(-x\log_{2}{10} = 2x\log_{2}{5}\), which led us to conclude that x = 0. 3. Verified the solution using a graphing calculator, observing that the graphs of \(y_1=10^{-x}\) and \(y_2=5^{2x}\) intersect at x = 0.

Step 1: Rewrite to a common base

To rewrite the equation using a common base, we note that both 10 and 5 can be expressed as powers of 2: \(10 = 2^{\log_{2}{10}}\) and \(5 = 2^{\log_{2}{5}}\). Plug in these expressions into the original equation: \((2^{\log_{2}{10}})^{-x} = (2^{\log_{2}{5}})^{2x}\) Now simplify:

Step 2: Equate the exponents and solve for x

Using the rules of exponents, we can simplify the equation further: \)2^{-x\log_{2}{10}} = 2^{2x\log_{2}{5}}\) Since both sides of the equation have the same base (2), we can set the exponents equal to each other: \[-x\log_{2}{10} = 2x\log_{2}{5}\] Now solve for x: \[-x\log_{2}{10} = 2x\log_{2}{5} \Rightarrow x(-\log_{2}{10}-2\log_{2}{5}) = 0\] Since \(\log_{2}{10}\) and \(\log_{2}{5}\) are nonzero, it must be the case that x = 0.

Step 3: Verification using a graphing calculator

To verify the solution using a graphing calculator, graph the two functions: 1. \(y_1=10^{-x}\) 2. \(y_2=5^{2x}\) Observe where the two functions intersect, which represents the solution to the equation. You should notice that at x=0, both functions intersect, confirming our algebraic solution above. Unfortunately, we cannot show the graph here, but using a graphing calculator or graphing software, you can visualize this result.

Key Concepts

Algebraic Solutions

Solving exponential equations algebraically involves isolating the variable of interest. It's important to manipulate the equation such that both sides have a common base, which simplifies comparison.
In the given problem, we have the exponential equation:

• \( 10^{-x} = 5^{2x} \)

To solve this algebraically, we aim to express both sides in terms of the same base (in this case, base 2). By doing this, we can exploit the properties of exponents that allow us to equate the exponents directly once the bases are identical. After expressing both 10 and 5 as powers of 2, we end up with:

• \( (2^{\log_{2}{10}})^{-x} = (2^{\log_{2}{5}})^{2x} \)

After simplification and application of the properties of exponents, we set the exponents equal:

• \[ -x\log_{2}{10} = 2x\log_{2}{5} \]
• Solve: \[ x(-\log_{2}{10} - 2\log_{2}{5}) = 0 \]

This equation implies that for x to be non-zero, the terms in the parenthesis must be zero. However, since they are not, x must equal zero to satisfy the equation.

Common Base Method

The common base method is a technique used to simplify and solve exponential equations. This method involves rewriting the bases of exponential terms to be the same.
This approach is useful when dealing with exponential terms that appear to be unrelated. In our case, both bases 10 and 5 can be converted to base 2.

Follow these steps:

• Identify bases that can potentially be expressed as powers of a common base.
• Rewrite each term in the equation using expressions with the common base. For example, 10 and 5 can be expressed using logarithms relative to base 2: \(10 = 2^{\log_{2}{10}}\) and \(5 = 2^{\log_{2}{5}}\).
• Substitute these expressions into the equation.
• Simplify by applying the properties of exponents (product, power of a power, etc.) until both sides share the same base.
• Set the exponents equal and solve for the unknown variable.

Utilizing a common base not only simplifies the algebraic manipulation, but also allows for straightforward comparison and resolution of the equation.

Graphing Calculator Verification

Verification of solutions using a graphing calculator is a handy visual method to confirm algebraic solutions. By graphing the functions on each side of the equation, we can visually inspect where they intersect.
For the equation \(10^{-x} = 5^{2x}\), follow these steps:

• Enter \( y_1 = 10^{-x} \) and \( y_2 = 5^{2x} \) into the graphing calculator.
• Set an appropriate viewing window to ensure you can capture potential intersections.
• Observe the graph and identify points where both graphs cross each other. These points represent the solutions to the equation.
• Confirm that these points match your algebraic solution. For this problem, we should expect the graphs to intersect at \(x = 0\), verifying our earlier work.

Using a graphing calculator provides a concrete visual confirmation which is particularly helpful to understand the behavior of exponential functions. It reinforces confidence in your solution and offers an illustrative check against algebraic errors.