Problem 83 - Calculus for Biology and Medicine

Question

Solve for \(x\). (a) \(\ln (x-3)=5\) (b) \(\ln (x+2)+\ln (x-2)=1\) (c) \(\log _{3} x^{2}-\log _{3} 2 x=2\)

Step-by-Step Solution

Answer

(a) \( x = e^5 + 3 \) or \( x \approx 151.413 \). (b) \( x \approx 2.592 \). (c) \( x = 18 \).

Step 1: Solve Equation (a)

We are given \( \ln(x - 3) = 5 \). To solve for \( x \), we first need to exponentiate both sides to remove the natural logarithm. This gives us \( x - 3 = e^5 \). Finally, isolate \( x \) by adding 3 to both sides, resulting in \( x = e^5 + 3 \).

Step 2: Simplify and Express as Decimal for (a)

Calculate \( e^5 \) using a calculator to find its approximate value. Since \( e \approx 2.718 \), \( e^5 \approx 148.413.\) Therefore, \(x \approx 148.413 + 3 = 151.413 \).

Step 3: Simplify Equation (b)

Consider the equation \( \ln(x + 2) + \ln(x - 2) = 1 \). Use the property of logarithms that states \( \ln(a) + \ln(b) = \ln(ab) \) to combine the logarithms. This gives us \( \ln((x+2)(x-2)) = 1 \).

Step 4: Exponentiate Equation (b)

Exponentiating both sides to eliminate the logarithm gives \((x+2)(x-2) = e^1 = e \). Expanding the left side, we find \( x^2 - 4 = e \).

Step 5: Solve Quadratic for (b)

Rewrite the equation as \( x^2 = e + 4 \). Therefore, solving for \( x \), we have \( x = \pm \sqrt{e + 4} \).

Step 6: Calculate Approximate Values for (b)

Using a calculator, find \( e \approx 2.718 \), so \( e + 4 \approx 6.718 \). Therefore, \( x \approx \sqrt{6.718} \). Calculating gives \( x \approx 2.592 \) or \( x \approx -2.592 \). Considering domain restrictions, only \( x \approx 2.592 \) is valid since \( x+2 \) and \( x-2 \) must be positive.

Step 7: Simplify Equation (c)

The equation \( \log_3(x^2) - \log_3(2x) = 2 \) can be combined using the properties of logarithms: \( \log_3\left(\frac{x^2}{2x}\right) = 2 \). This simplifies to \( \log_3\left(\frac{x}{2}\right) = 2 \).

Step 8: Exponentiate Equation (c)

Exponentiate both sides to eliminate the logarithm, resulting in \( \frac{x}{2} = 3^2 \). Simplify to find \( \frac{x}{2} = 9 \).

Step 9: Solve for x in Equation (c)

Multiply both sides by 2 to isolate \( x \). This gives \( x = 18 \).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm with the base \( e \). Here, \( e \) is an irrational and transcendental number approximately equal to 2.718. The natural logarithm is commonly used in mathematics, especially in contexts involving growth processes, such as population growth, and decay processes, like radioactive decay. To solve equations involving \( \ln \), you often need to perform a process called exponentiation. This involves raising \( e \) to the power of the logarithm expression. For instance, if you have an equation like \( \ln(x-3) = 5 \), you can remove the logarithm by rewriting it as:\[ x - 3 = e^5 \]This transformation is based on the definition of a logarithm: \( b^y = x \) if and only if \( \log_b x = y \). In our case, \( b=e \) and \( y = 5 \).The natural logarithm is a fundamental concept that serves as the basis for understanding and solving logarithmic equations effectively.

Properties of Logarithms

The properties of logarithms are essential tools for simplifying and solving logarithmic equations. These properties allow us to combine, separate, and manipulate logarithmic expressions based on established rules. Understanding these properties is crucial when working with multiple logarithmic terms.One key property is the Product Rule, which states that \( \log_b(M \times N) = \log_b M + \log_b N \). This property is useful when you need to combine logarithms. For instance, in the equation \( \ln(x+2) + \ln(x-2) \), you can use the product rule to write it as:\[ \ln((x+2)(x-2)) \]This simplifies the original equation, making it easier to solve.Another important property is the Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \). This property helps when dividing logarithmic expressions, as used in the equation \( \log_3(x^2) - \log_3(2x) \), simplifying to:\[ \log_3\left(\frac{x^2}{2x}\right) = 2 \]These properties transform complex expressions into manageable forms, allowing you to solve logarithmic equations step by step.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving these equations often involves finding the values of \( x \) that make the equation true.One common method of solving quadratic equations is to isolate \( x^2 \) and take the square root of both sides. For example, if you have an equation like \( x^2 - 4 = e \), you can rewrite it as:\[ x^2 = e + 4 \]Taking the square root gives:\[ x = \pm \sqrt{e + 4} \]The solution involves checking for valid solutions within the context of the problem, especially when dealing with logarithmic domains, as seen in the original logarithmic equation where only positive \( x \) values make sense. Quadratic equations often yield two solutions due to the square root’s nature, and understanding these solutions helps in effectively solving a variety of mathematical problems.