Problem 32 - 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. $$30-3(0.75)^{x-1}=29$$

Step-by-Step Solution

Answer

The exact solution is \(x = 4.338863...\), which approximates to \(x \approx 4.339\).

Step 1: Isolate the Exponential Term

Start by isolating the exponential term \(3(0.75)^{x-1}\) on one side of the equation. Subtract 29 from both sides:\[30 - 29 = 3(0.75)^{x-1}\]This simplifies to:\[1 = 3(0.75)^{x-1}\]

Step 2: Divide by Coefficient

Divide both sides of the equation by 3 to isolate \((0.75)^{x-1}\):\[ \frac{1}{3} = (0.75)^{x-1} \]

Step 3: Apply the Logarithm

Take the natural logarithm of both sides to solve for \(x-1\):\[ \ln\left(\frac{1}{3}\right) = \ln\left((0.75)^{x-1}\right) \]Using the logarithmic identity \(\ln(a^b) = b\ln(a)\), this becomes:\[ \ln\left(\frac{1}{3}\right) = (x-1)\ln(0.75)\]

Step 4: Isolate x

Divide both sides by \(\ln(0.75)\) to solve for \(x - 1\):\[ x-1 = \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.75)} \]Solve for \(x\):\[ x = 1 + \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.75)} \]

Step 5: Calculate the Exact Solution

Using a calculator, find the value of the expression:\[ x = 1 + \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.75)} \]This gives the exact solution for \(x\).

Step 6: Approximate to Nearest Thousandth

Using a calculator, approximate \(x\) to the nearest thousandth. After calculating, we get:\[ x \approx 4.339 \]

Key Concepts

Exact Form Solutions

Exact form solutions involve solving equations exactly using algebraic manipulations rather than decimal approximations.
This type of solution provides a more precise understanding without rounding off intermediate steps.
For the given problem, we start by isolating the exponential term:

• Subtract 29 from both sides, yielding: \( 30 - 29 = 3(0.75)^{x-1} \)
• Simplify to get \( 1 = 3(0.75)^{x-1} \)
• Next, divide by 3: \( \frac{1}{3} = (0.75)^{x-1} \)

To solve this, we apply logarithms to both sides. This exact expression uses logarithmic identities to find \( x \) without rounding off any values until the final answer.
Taking logs gives us an exact equation involving a fraction and logarithms, preserving the full precision of the solution as: \[ x = 1 + \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.75)} \] This form provides an exact solution that reflects the precise mathematical relationship.

Logarithmic Identities

Logarithmic identities are essential for transforming and solving exponential equations like this one.
These identities simplify the complexities involved when dealing with powers and roots. Important identities used in this problem include:

• \( \ln(a^b) = b \cdot \ln(a) \), which turns the exponent into a coefficient.
• \( \ln(a) - \ln(b) = \ln(\frac{a}{b}) \), useful in simplifying expressions derived from division.

In our exercise, after isolating the exponential term \( (0.75)^{x-1} = \frac{1}{3} \), we took the natural logarithm:\[ \ln\left(\frac{1}{3}\right) = (x-1) \cdot \ln(0.75) \]Using these identities allows us to isolate \( x \) by transforming the complex fractional expressions into simpler linear algebraic equations that are easier to handle.

Numerical Approximation

Numerical approximation comes into play when you need a practical, rounded-off solution.
This process involves using a calculator to evaluate expressions and reach an approximate answer.In this problem, once we had our exact expression for \( x \), we used a calculator to find the decimal value:

• The expression \( x = 1 + \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.75)} \) requires evaluating natural logarithms and performing division.
• Using a scientific calculator, we plug these values to find that \( x \approx 4.339 \), an approximation to the nearest thousandth.

This approximation is useful when an exact form is unnecessary within a practical context or when quickly communicating the results without needing full precision.

Calculator Usage in Math

Calculators are invaluable tools in both validating solutions and finding numerical approximations. They facilitate complex calculations that involve logarithms, multiplication by powers, and division of logarithmic results.In this instance, a calculator helps to evaluate:

• Logarithmic expressions such as \( \ln\left(\frac{1}{3}\right) \) and \( \ln(0.75) \).
• Final division \( \frac{\ln\left(\frac{1}{3}\right)}{\ln(0.75)} \) to find the precise value of \( x-1 \).

When approximating to the nearest thousandth, using a calculator ensures that all steps are accurate and consistent. It also helps in cross-verifying different methods of solving exponential equations, giving both students and educators confidence in the results.