Problem 22 - Algebra 2

Question

Use a table to solve each equation. Round to the nearest hundredth. $$ 2^{2 x-1}=3^{x} $$

Step-by-Step Solution

Answer

The solution for the equation is \(x \approx \frac{1}{2 - log_{2}(3)}\).

Step 1: Express the equation to be able to use logarithmic properties

Rewrite \(2^{2x-1}\) as \(2^{2x} / 2\). This manipulation allows to isolate exponentials, leading to simpler manipulations. So the equation becomes \(2^{2x} / 2 = 3^x\).

Step 2: Logarithmic conversion

Take the logarithm base 2 on both sides. This will give us \((2x)log_{2}(2) - log_{2}(2) = x log_{2}(3)\).

Step 3: Simplification

Simplify the equation by using logarithmic rules. It will convert to \(2x - 1 = x log_{2}(3)\).

Step 4: Solve for x

Isolate x by taking the common factor out. This gives x(2 - log_{2}(3)) = 1. Solving for x, we get \(x = \frac{1}{2 - log_{2}(3)}\).

Step 5: Calculate x value.

Compute using a calculator to find the approximate value of x. Keep the precision to the nearest hundredth.

Key Concepts

Logarithmic Properties

When dealing with exponential equations, logarithmic properties become a powerful tool. These properties allow us to manipulate and simplify complex expressions. The basic property of logarithms is that they are the inverse operations of exponents. This means if you have an exponential equation, you can use logarithms to "undo" the exponent.

For example, if you have an equation like \(a^x = b^y\), using logarithmic properties, such as the change of base formula or the power rule, you can convert this to \(x \log(a) = y \log(b)\). Here, you are essentially bringing down the exponent as a multiplier, which leads to more manageable arithmetic operations. This process drastically simplifies solving the equation.

These properties also allow us to choose different bases for logarithms, such as the natural logarithm (\(\ln(x)\)) or the common logarithm (\(\log_{10}(x)\)). This flexibility is especially important in solving real-world problems or equations, like in our original exercise. Using logarithms helps in isolating variables and simplifying the expression into something more easily solvable. Remember, practice with various logarithmic properties can make them much more intuitive over time.

Rounding to the Nearest Hundredth

Rounding to the nearest hundredth is a specific form of simplifying a number to make it more understandable or comparable. This is especially relevant when dealing with decimals or when a exact value is not necessary. In mathematics and applied sciences, results often need to be rounded to give a clearer picture or to fit within given constraints.

To round a number to the nearest hundredth, follow this simple approach:

• Identify the digit in the hundredth place, which is the second digit to the right of the decimal point.
• Look at the third digit to the right of the decimal. If this digit is 5 or higher, increase the hundredth place digit by one.
• If the third digit is less than 5, leave the hundredth place digit as is.

This rounding method provides an approximation that is close to the original value, making it suitable for quick estimations or when validation to two decimal places is sufficient. In our solved problem, after computing the value of x, rounding to the nearest hundredth helps finalize the answer in a simple and readable format.

Table Method for Solving Equations

The table method for solving equations is a strategic approach used when traditional algebraic methods might be cumbersome. This technique involves evaluating both sides of an equation over a range of values and observing where the two sides are equal. It's particularly useful when dealing with non-linear equations, such as exponential equations, where solutions might not be straightforward.

To employ the table method:

• Create a table with two columns, one for each side of the equation.
• Select a range of input values for your variable (often x) and calculate each corresponding side of the equation.
• Identify when the values in both columns are equivalent or nearly equivalent.

The beauty of the table method is its visual nature. Observing where the values match allows for a clear view of the solution, aiding intuition and understanding. Although this method can be time-consuming, it provides a useful alternative for complex equations where analytical solutions are difficult to achieve. It can also complement analytical techniques by providing a preliminary check of expected solutions.