Problem 106 - 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. $$\ln \sqrt{x-8}=5$$

Step-by-Step Solution

Answer

The solution to the equation \(\ln \sqrt{x-8}=5\) rounded to three decimal places is \(x=22034.47\).

Step 1: Analyze and simplify the equation

First, break down the square root using the property that \(\sqrt[n]{a}=a^{1/n}\). Thus the given equation \(\ln \sqrt{x-8}=5\) becomes \(\ln (x-8)^{1/2}=5\).

Step 2: Turning logarithmic equation into exponential

Using the property that turns a logarithm to an exponent \(b=\ln a \rightarrow e^{b}=a\), the equation becomes \(e^{5}=(x-8)^{1/2}\).

Step 3: Solve for x

Next, we can eliminate the square root by squaring both sides of the equation which gives \((e^{5})^{2}=(x-8)\). Simplifying leads to \(e^{10}=x-8\). Once we solve for \(x\), \(x=e^{10}+8\). Given that \(e^{10} \approx 22026.47\), adding 8 to both sides to solve for \(x\) gives approximately \(22034.47\).

Step 4: Check for extraneous solutions

Logarithms are undefined for values less than or equal to zero. Therefore substituting \(x\) in the original equation, \(\ln \sqrt{x-8}\), we get \(\ln \sqrt{22034.47-8} = \ln \sqrt{22026.47}\) which gives approximately a value of \(5\), thus verifying the solution.

Key Concepts

Natural Logarithms

Natural logarithms are a fundamental concept in mathematics, represented using the symbol \( \ln \). They are logarithms to the base \( e \), where \( e \approx 2.71828 \) is an irrational and transcendental number. Natural logarithms are used extensively in calculus and mathematical modeling.

• A natural logarithm represents the power to which the base \( e \) must be raised to obtain a certain number.
• For example, \( \ln e = 1 \) because \( e^1 = e \).
• Natural logs are particularly useful in solving exponential equations.

When solving equations involving natural logarithms, it's important to remember their properties, such as turning the logarithmic expression into an exponential one. This transformation allows for easier manipulation when solving for the unknown variable.

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They can model growth and decay processes in nature, such as population growth, radioactive decay, and interest calculations.

• The general form is \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants.
• In the context of solving logarithmic equations, exponentiation is the inverse operation of taking the logarithm.

In the given exercise, we encountered an exponential equation derived from a logarithmic equation: \( e^5 = (x-8)^{1/2} \). This transformation allows us to move away from logarithms and use exponential functions to solve for \( x \).By squaring both sides of the resulting equation, we remove the square root and further simplify the equation: \( (e^5)^2 = x-8 \). This step is crucial as it enables us to solve for the variable by isolating \( x \) on one side of the equation.

Solving Equations

Solving logarithmic equations involves a series of mathematical operations to isolate the variable. In this exercise, we started with the equation \( \ln \sqrt{x-8} = 5 \) and transformed it into an exponential equation. Key steps involved in solving the equation included:

• Using properties of logarithms to express the equation in a simpler form.
• Converting the logarithmic equation into an exponential form.
• Squaring both sides to eliminate the square root and solve for the variable \( x \).

The solution involved finding \( e^{10} \approx 22026.47 \), and thus \( x \approx 22034.47 \), after verifying the calculations. Always remember to check your solution for extraneous results. In this case, substituting back into the original equation confirmed the correctness since the derived \( x \) value satisfies the equation, which means the solution is valid.

Graphing Utilities

Graphing utilities, such as graphing calculators or computer software, are indispensable tools for verifying solutions to mathematical equations visually.

• They help in checking the accuracy of our solutions derived algebraically.
• Visualization of functions can reveal more about their nature, like intersections points or asymptotic behavior.

In this exercise, after solving \( \ln \sqrt{x-8} = 5 \) algebraically, we used graphing utilities to confirm the solution. By graphing the function \( y = \ln \sqrt{x-8} \) and the horizontal line \( y = 5 \), we can visually verify the point where these graphs intersect, ensuring that \( x \approx 22034.47 \) is indeed a valid solution.This dual approach of algebraic and visual verification enhances our understanding and provides confidence in our solution.