Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$-2+2 \ln 3 x=17$$

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 1325.808\). After graphing the equation and the line, one can verify that the solution is correct.

Step 1: Isolate the logarithm

First, isolate the logarithm on one side of the equation by adding 2 to both sides to get \(2 \ln 3x = 17 + 2\), which simplifies to \(2 \ln 3x = 19.\)

Step 2: Remove the coefficient of the logarithm

Next, divide by 2 on both sides to get \(\ln 3x = 19 / 2\), which simplifies to \(\ln 3x = 9.5\).

Step 3: Use exponentiation

Rewrite the logarithmic equation in exponential form. This gives the equation \(e^{9.5} = 3x\).

Step 4: Solve for x

Now, solve for x by dividing both sides by 3. This gives \(x = \frac{e^{9.5}}{3}\). Using a calculator, compute the value of x to three decimal places to find \(x \approx 1325.808\).

Step 5: Verify solution graphically

Finally, verify the solution graphically. On a graphing utility, graph both the original equation and the line \(x = 1325.808\). If the line and the curve intersect, the solution is verified.

Key Concepts

Solving Logarithmic Equations Algebraically

Understanding how to solve a logarithmic equation algebraically is fundamental in algebra. When you encounter a problem like \( -2+2 \ln 3 x=17 \), the first step is to gather all the logarithmic parts on one side. In this case, add 2 to both sides to isolate the logarithmic term, obtaining \(2 \ln 3x = 19\).

Next, you need to remove any coefficients from the logarithm, so you divide by 2, leading to \(\ln 3x = 9.5\). Remember to perform the same operations on both sides of the equation to maintain balance. After simplifying the logarithmic part, you're one step closer to finding the value of \(x\).

Exponential Form Equation

Converting a logarithm to its exponential form is essential in solving logarithmic equations. For the equation \(\ln3x=9.5\), we utilize the property that \( \ln a = b \) is equivalent to \( e^b = a\).

Using this concept, \(\ln 3x = 9.5\) transforms to \(e^{9.5} = 3x\). This step takes the equation out of its logarithmic form and presents it as an exponential equation, which is typically easier to solve. Once you've rewritten in exponential form, you're now ready to isolate \(x\) and solve for its value.

Isolation of Logarithm

To clearly see the solution of the variable within a logarithmic equation, isolating the logarithm is a crucial step. We started with \(2 \ln 3x = 19\) and divided both sides by 2 to get \(\ln 3x = 9.5\), thus isolating \(\ln 3x\).

Isolating the logarithm simplifies the equation, preparing it to be rewritten in exponential form. By doing this, we separate the \(x\) from other elements in the equation, allowing for direct solving or further manipulation.

Graphical Verification of Solutions

After algebraically solving a logarithmic equation and arriving at a solution like \(x \text{ is approximately } 1325.808\), it's prudent to confirm this result graphically. Use a graphing tool to plot the original equation \( -2+2 \ln 3 x=17\) and draw a line at the solved value of \(x\), in this case, \(x = 1325.808\).

If both the curve of the original equation and the line intersect at \(1325.808\), then the algebraic solution is correct. Graphical verification is a powerful tool for validating solutions, especially when equations become complex or have multiple valid answers.