Problem 68 - A Graphical Approach to College Algebra

Question

Use a graphing calculator to find the solution of each equation. Round your result to the nearest thousandth. $$1.5^{\ln x}=10^{0.5}$$

Step-by-Step Solution

Answer

The solution is approximately \(x \approx 13.132\).

Step 1: Rewrite the Equation

The given equation is \(1.5^{\ln x} = 10^{0.5}\). First, convert the expression on the right side into a more manageable number. Notice that \(10^{0.5} = \sqrt{10}\), which is approximately 3.162.

Step 2: Simplify the Expression

We simplify the equation to \(1.5^{\ln x} = 3.162\). Next, take the natural logarithm (ln) of both sides to facilitate solving for \(x\): \(\ln(1.5^{\ln x}) = \ln(3.162)\).

Step 3: Apply Logarithmic Properties

Apply the power rule of logarithms: \(\ln(a^b) = b\ln(a)\). This simplifies the left side to \(\ln x \cdot \ln(1.5)\). The equation now looks like \(\ln x \cdot \ln(1.5) = \ln(3.162)\).

Step 4: Solve for \(\ln x\)

Divide both sides by \(\ln(1.5)\) to isolate \(\ln x\): \(\ln x = \frac{\ln(3.162)}{\ln(1.5)}\).

Step 5: Solve for \(x\)

To find \(x\), exponentiate both sides of the equation: \(x = e^{\frac{\ln(3.162)}{\ln(1.5)}}\). Use a calculator to compute this value.

Step 6: Calculate using the Calculator

Use a graphing calculator to find \(x\). First, calculate \(\ln(3.162)\) and \(\ln(1.5)\), then perform division followed by exponentiation as in the previous step. Finally, round the result to the nearest thousandth.

Key Concepts

Using a Graphing Calculator Effectively

A graphing calculator is a powerful tool that helps visualize and solve equations. When tackling a problem like \(1.5^{\ln x} = 10^{0.5}\), the graphing calculator can quickly plot the functions and identify intersections. Here's how you can use it:

• Start by graphing each side of the equation separately. Enter \(y_1 = 1.5^{\ln x}\) and \(y_2 = 10^{0.5}\).
• Observe where these graphs intersect. This point represents the solution to the equation.
• Zoom in or adjust the view settings to pinpoint the exact value of \(x\).

Utilizing this tool saves time and increases accuracy in solving complex mathematical equations.

Basics of Equation Solving

Equation solving involves finding the unknown value that satisfies an equation. In the given problem, you're solving for \(x\) in \(1.5^{\ln x} = \sqrt{10}\). The key steps include:

• Transforming complex expressions into simpler forms using mathematical properties.
• Applying logarithmic rules like \(\ln(a^b) = b\ln(a)\) to enable algebraic manipulation.
• Isolating the variable, in this case \(\ln x\), to make solving straightforward.

These techniques systematically break down the problem, leading to a solution step by step.

Understanding the Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), a mathematical constant approximately equal to 2.718. It simplifies expressions involving exponential growth and decay. In the equation \(1.5^{\ln x} = 3.162\):

• We apply the natural logarithm to both sides, reducing the complexity of the expression.
• Using \(\ln\) helps in transforming power equations into a format where algebraic methods can solve for the variable.
• Knowing \(e^{\ln(x)} = x\) is crucial, as it provides a direct way to reverse the logarithm operation to solve for \(x\).

Understanding \(\ln\) and its properties is essential for solving a wide range of mathematical problems.

Exploring Logarithmic Properties

Logarithmic properties are rules that simplify expressions and equations involving logarithms. These are vital in solving equations like \(1.5^{\ln x} = 10^{0.5}\). Here are the key properties used:

• The Power Rule: \(\ln(a^b) = b\ln(a)\), which helps convert powers into products.
• The Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\), useful for breaking down products into sums.
• The Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\), for simplifying divisions into subtractions.

By mastering these rules, you can easily manipulate logarithmic equations, making them simpler to solve.