Problem 14 - Algebra 2

Question

Solve by graphing. Round to the nearest ten-thousandth. $$ 5^{3 x}=500 $$

Step-by-Step Solution

Answer

To yield the final result, the computed value of 'x' should be rounded to the nearest ten-thousandth. The exact value depends on the outcome of the graph.

Step 1: Convert Equation to Graphing Form

The given equation is \(5^{3x}=500\). First, rewrite the equation so it is suitable for graphing. This can be accomplished by taking the natural logarithm (ln) of both sides of the equation to get \(3x = ln(500)/ln(5)\). Then, isolate 'x' by dividing both sides by 3 to obtain \(x = ln(500)/(3ln(5))\).

Step 2: Graph the Function

Then, graph the function \(y = ln(x)/(3ln(5))\) using graphing tools. Identify the x-coordinate of the point where the curve intersects with the line \(y=500\).

Step 3: Round to the Nearest Ten-Thousandth

The obtained x-coordinate represents the value of 'x' that solves the original equation. But, as per the problem requirements, round this value to the nearest ten-thousandth place.

Step 4: Conclude Solution

The rounded value of 'x' is the solution to the given equation. If this value is plugged into the initial equation at the place of 'x', the left and right sides would be approximately equal, confirming the obtained 'x' is the solution to the given equation.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions is an effective way to visualize solutions to equations like \(5^{3x} = 500\). It allows us to see where our function intersects certain values. The critical point on the graph is where the equation \(y = 5^{3x}\) meets the horizontal line \(y=500\). This visual approach can offer a clear, intuitive understanding of the approximate solution by observing the graph’s behavior at intersections.When graphing:

• Plot the function of the form \(y = 5^{3x}\).
• Draw a horizontal line at \(y=500\).
• The x-coordinate of the intersection point is the approximate solution.

Using graphing technology, you can quickly and accurately locate this intersection point, providing a visual confirmation of your solution before continuing with algebraic techniques.

Understanding Logarithms in Equations

Logarithms are a powerful tool for solving equations involving exponents, such as \(5^{3x} = 500\). They allow us to work with the exponents in a more manageable form. By taking the natural logarithm (ln) of both sides, the equation simplifies into: \(3x = \frac{\ln(500)}{\ln(5)}\).Here's why logarithms are used:

• Transformations: Converting from exponential to logarithmic form makes the variable "x" easier to isolate.
• Simplification: Logarithms turn multiplicative processes into additive ones, simplifying calculations significantly.

Once you understand the basic properties, like \(\ln(a^b) = b \times \ln(a)\), logarithms become a straightforward method for solving exponential equations.

Solving Exponential Equations

Exponential equations, like \(5^{3x} = 500\), can initially seem complex due to the exponential variable. However, by applying the right techniques, they become solvable.To solve such equations:

• Isolate the Exponential: Separate the exponential expression on one side of the equation.
• Utilize Logarithms: Apply logarithms to both sides to eliminate the exponential term.
• Solve for the Variable: Rearrange to isolate "x”, yielding a formula involving logarithms that can be calculated.

By following these steps, the original equation \(5^{3x} = 500\) can be efficiently transformed into a form that allows clear and straightforward calculation of the variable "x".

Rounding Numbers for Precision

Rounding numbers is crucial when dealing with mathematical solutions, especially when instructed to round to specific decimal places. Rounding to the nearest ten-thousandth means ensuring that the final solution is both accurate and matches the precision requested.To round correctly:

• Identify the Decimal Place: Look at the fifth decimal place to determine how the fourth place should be adjusted.
• Apply Rounding Rules: If the fifth decimal is 5 or greater, round up the fourth number.
• Carry Over Adjustments: Ensure rounding does not affect the overall accuracy of the solution.

By rounding correctly, your calculated solution \(x\) remains reliable and adheres to the precision required, providing a clear and trustworthy result.