Problem 28 - A Graphical Approach to College Algebra

Question

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$5(1.2)^{3 x-2}+1=11$$

Step-by-Step Solution

Answer

Exact: \(x = \frac{\ln(2)}{3\ln(1.2)} + \frac{2}{3}\); Approximate: \(x \approx 1.569\).

Step 1: Isolate the exponential term

Start by subtracting 1 from both sides of the equation to begin isolating the exponential expression. The equation becomes:\[5(1.2)^{3x-2} = 10\]

Step 2: Divide to further isolate the exponential term

Divide both sides of the equation by 5 to completely isolate the exponential term:\[(1.2)^{3x-2} = 2\]

Step 3: Apply logarithms

Apply the natural logarithm (ln) to both sides of the equation to facilitate solving for \(x\). This gives:\[\ln((1.2)^{3x-2}) = \ln(2)\]

Step 4: Use logarithmic identity

Apply the power rule of logarithms, \(\ln(a^b) = b\ln(a)\), to simplify the left side:\[(3x-2)\ln(1.2) = \ln(2)\]

Step 5: Solve for x

First, solve for \(3x-2)\) by dividing both sides by \(\ln(1.2)\):\[3x-2 = \frac{\ln(2)}{\ln(1.2)}\]Next, add 2 to both sides:\[3x = \frac{\ln(2)}{\ln(1.2)} + 2\]Finally, divide everything by 3 to solve for \(x\):\[x = \frac{\ln(2)}{3\ln(1.2)} + \frac{2}{3}\]

Step 6: Calculate the exact solution (if possible)

Using a calculator, compute the value of \(x\) exactly by plugging in the natural log computations:\[x = \frac{\ln(2)}{3\ln(1.2)} + \frac{2}{3}\]

Step 7: Approximate x to the nearest thousandth

Use a calculator to find the numerical value:\[x \approx \frac{\ln(2)}{3\ln(1.2)} + \frac{2}{3} \approx 1.569\]

Key Concepts

Isolation of Exponential Expressions

To solve an exponential equation, the first step is to isolate the term that contains the exponent. This means getting rid of any additional numbers or terms that are affecting the exponential component. In our example, the equation begins as:

• \(5(1.2)^{3x-2} + 1 = 11\)

The goal is to have the exponential expression \((1.2)^{3x-2}\) by itself on one side of the equation. We start by subtracting 1 from both sides, moving one step closer to isolation:

• \(5(1.2)^{3x-2} = 10\)

Next, divide both sides by 5 to completely isolate:

• \((1.2)^{3x-2} = 2\)

This separation is crucial, as it allows us to focus on the exponential term, preparing it for further operations, like applying logarithms.

Natural Logarithms

Natural logarithms, denoted as \(\ln\), are logarithms with base \(e\), a special irrational number approximately equal to 2.71828. In mathematical practices, natural logarithms provide a powerful tool in solving exponential equations, especially those involving the constant \(e\). They convert multiplicative processes into additive processes, making calculations significantly simpler. In our example:

• \(\ln((1.2)^{3x-2}) = \ln(2)\)

By applying \(\ln\) to both sides, we utilize the properties of logarithms, particularly the power rule, which helps unravel the exponent, allowing us to solve for \(x\). This transformation is what makes logarithms so useful in these kinds of equations.

Solving Equations with Logarithms

Once the exponential term is isolated and the natural logarithm applied, solving the equation involves using properties of logarithms to simplify and solve for the variable. In our problem, we use the logarithm power rule, \(\ln(a^b) = b \ln(a)\), to bring down the exponent in front:

• \((3x-2) \ln(1.2) = \ln(2)\)

This transforms the equation into a linear form with respect to \(x\). We then isolate \(x\) by:

• Dividing both sides by \(\ln(1.2)\)
• Adding 2 to the equation
• Dividing everything by 3

Following these steps provides a precise formula for \(x\) in terms of natural logarithms.

Approximation of Irrational Numbers

After finding the solution in exact form, it's often necessary to approximate irrational numbers to a practical degree of accuracy. This is particularly useful when an exact value can't be easily used in real-world applications. In mathematics, we approximate to a specified number of decimal places for clarity and usability.
Using a calculator, we compute the final result for \(x\) obtained from:

• \(x = \frac{\ln(2)}{3\ln(1.2)} + \frac{2}{3}\)

By calculating this expression, we get an approximate value:

• \(x \approx 1.569\)

Here, we've rounded \(x\) to the nearest thousandth place, which balances precision with practicality, providing a clear, concise answer.