Problem 65 - Algebra and Trigonometry

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln \sqrt{x+3}=1 $$

Step-by-Step Solution

Answer

The exact solution is \(x = e - 3\). The approximate solution, to two decimal places, is \(x = -0.282\).

Step 1: Exponentiate both sides

Exponentiate both sides of the equation using the base of the natural logarithm, which is e. This transforms the equation into the form \(x+3\) is equal to \(e^1\). So the new equation is: \(x + 3 = e\).

Step 2: Subtract 3 from both sides

Subtract 3 from both sides to isolate \(x\) on one side of the equation, giving the equation as: \(x = e - 3\).

Step 3: Check the domain

Verify that the solution is within the domain of the original logarithmic function, \(\ln \sqrt{x+3}\). This requires ensuring that \(x+3\) is positive (since we cannot square root a negative number) and \(x >= -3\). In this case, \(e - 3\) is greater than \(-3\) , so the solution fits into the domain.

Step 4: Approximation

To get an approximation of the solution correct to two decimal places, solve \(x= e-3\) using a calculator. Note that e is approximately 2.718. After subtracting 3 from e, we get \(x \approx -0.282\).

Key Concepts

Understanding the Natural Logarithm

Natural logarithms have a base of Euler's number, commonly represented as \(e\). Euler's number is an irrational constant approximately equal to 2.718. The natural logarithm function, denoted as \(\ln(x)\), is the inverse operation to exponentiation with base \(e\).

• If you have \(\ln(x) = y\), it implies that \(e^y = x\).
• This means natural logarithms help in solving equations where the variable is in the exponent.

In the given exercise, the natural logarithm function is used to handle a square root expression. By understanding the natural logarithm, we can rewrite the equation in linear form and solve for \(x\). This is useful because it simplifies potentially complex logarithmic equations.

Interpreting Domain Restrictions

The domain of a logarithmic function refers to the set of all possible input values (\(x\)) for which the logarithm is defined. For logarithms involving square roots, special considerations are needed. The argument of the square root must be non-negative, and thus:- For \(\ln(\sqrt{x+3})\), we require \(x+3 > 0\).- Therefore, \(x\) must be greater than or equal to \(-3\).These restrictions ensure that we do not attempt to take the natural log of a non-positive number, as logarithms for non-positive values are undefined. In solving the equation \(x + 3 = e\), it confirms that \(x = e - 3\) fits within the permissible domain.

Obtaining the Exact Solution

An exact solution means we provide the answer in terms of known constants and mathematical operations, rather than approximating numerically. By following our step-by-step process:1. We exponentiate both sides of the equation using base \(e\) to remove the logarithm. This step is critical as it converts the equation into a simpler form.2. We isolate \(x\) by subtracting 3 from both sides, resulting in \(x = e - 3\).This manipulation gives us the exact solution in terms of \(e\), which is a standard way to express solutions involving natural logarithms because \(e\) is a well-known mathematical constant.

Understanding Decimal Approximation

Decimal approximation is used when solutions cannot be easily expressed as a simple number, which is common with irrational numbers like \(e\). To obtain a useful approximation:- Use a calculator to compute \(e\), noticing \(e \approx 2.718\).- Substitute this approximation into the expression for \(x\), yielding \(x \approx 2.718 - 3\).This calculation gives \(x \approx -0.282\). Decimal approximations are important because they provide a practical number that can be more easily interpreted or utilized in real-world applications.It's fundamental to round your answer to a required number of decimal places, in this case, two decimal places, to ensure the precision in applications.