Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$3 \ln 5 x=10$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 4 & 5 & 6 & 7 & 8 \\\\\hline 3 \ln 5 x & & & & & \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

From the table, it appears that the solution lies in the interval (6, 7). Graphically, it also shows the solution approximates to this interval. Solving algebraically, the solution is approximately \(x = 6.685\).

Step 1: Completing the Table

In this step, substitute the provided x-values (4, 5, 6, 7, 8) into the equation \(3 \ln 5x\), round off your results to three decimal places, and fill up the table. This will give an estimate of the interval containing the solution.

Step 2: Estimate the Solution Using Graph

Graphing helps in estimating the solution. Plot the function \(3 \ln 5x\) and the constant 10 on the same graph. The x-coordinate of the point of intersection would be the solution of the equation.

Step 3: Solving the Equation Algebraically

In this step, solve the equation algebraically for a precise solution. Start by isolating the logarithm by dividing both sides by 3: \(\ln 5x = \frac{10}{3}\). Then, convert the logarithm to the corresponding exponential form: \(5x = e^{10/3}\). Lastly, solve for x by dividing both sides by 5: \(x = \frac{e^{10/3}}{5}\). Round your results to three decimal places.

Key Concepts

Graphing Utilities

Graphing utilities are valuable tools for visualizing mathematical equations, especially when searching for solutions. For this exercise, using a graphing utility means plotting the equation \( 3 \ln 5x \) and the horizontal line \( y = 10 \). The point where these graphs intersect represents the solution where both functions have the same value for \( x \).

By plotting both these equations, you will visually see how the curve and the line meet at a particular point along the x-axis. This visually indicates the approximate value of \( x \) that satisfies the equation. If you use graphing software or a calculator, it often provides tools to identify this intersection point as a coordinate.

• The x-coordinate of this intersection point is the approximate solution to your equation.
• Graphing also reveals the trend of your logarithmic function and how it behaves compared to a linear constant.

This estimation method is valuable because it gives a quick overview before you dive into algebraic methods that offer precise calculations.

Interval Estimation

Interval estimation involves identifying a range of values for \( x \) that contains the solution to an equation. This task means evaluating the values of \( 3 \ln 5x \) at given points (e.g., \( x = 4, 5, 6, 7, 8 \)) and checking where the output is close to the constant, in this case, 10.

Start by substituting each \( x \)-value into \( 3 \ln 5x \), then calculating and rounding to three decimal places. Compare these results with 10:

• If the result is less than 10, you know the solution lies at a higher x-value.
• If the result is greater than 10, then it lies lower than this x-value.

Continuing this way gives you a range where the solution might be located. The interval gives a confined scope to later apply a more precise algebraic solution. Interval estimation helps use logical reasoning to determine where solutions lie even without precise tools.

Algebraic Solutions

Solving logarithmic equations algebraically allows for precise solutions in comparison to graph-based estimations. In our case, the solution to the equation \(3 \ln 5x = 10\) requires isolating the logarithmic part of the equation.

First, divide both sides by 3 to simplify the logarithm: \(\ln 5x = \frac{10}{3}\). Then, solve the logarithm equation by transforming it to an exponential format. This format, \( 5x = e^{\frac{10}{3}} \), is derived from the property that if \( \ln a = b \), then \( a = e^b \).

• Finally, solve for \( x \) by dividing by 5: \(x = \frac{e^{\frac{10}{3}}}{5}\).
• This provides the exact value of \( x \), which can then be rounded to three decimal places.

Using algebraic solutions is key to finding exact numbers which help confirm the approximate values obtained via graphical methods.