Problem 82 - College Algebra and Calculus: An Applied Approach

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln (x+1)-\ln (x-2)=\ln x$$\ln (x+1)-\ln (x-2)=\ln x\)

Step-by-Step Solution

Answer

The solution to the equation is \(x=3.30278\).

Step 1: Apply Logarithmic Properties

Firstly, use the property of logarithms \(\ln a - \ln b = \ln (a/b)\) to simplify the left side of the equation. Apply this to get \(\ln \frac{(x+1)}{(x-2)}=\ln x\)

Step 2: Clear Logarithms

In a scenario where the log of two expressions are equal, those expressions themselves must be equal. So, \(\frac{(x+1)}{(x-2)}=x\)

Step 3: Formulate Quadratic Equation

The equation from Step 2 is a fractional equation. Multiply everything by \(x-2\) to clear the fraction. This yields a quadratic equation, \((x+1)=x(x-2)\)

Step 4: Solve the Quadratic Equation

Simplify and solve the equation to get \(x^2-3x-1=0\). This can be solved either by factoring, completing the square, or applying the quadratic formula.

Step 5: Apply the Quadratic Formula

This equation is not easily factored, so the quadratic formula is employed. Recall that the quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Thus, the roots are \(x=\frac{3 \pm \sqrt{9+4}}{2}\).

Step 6: Compute roots of Quadratic Equation

Upon simplification, the roots are \(x=3.30278\) and \(x=-0.30278\)

Step 7: Check domain of roots

Because the given equation involves logarithms, the solutions must be in the domain of the original logarithmic expressions. Logarithms are only defined for positive numbers. Check if the roots are in the domains. Reject -0.30278 because it's negative. Check 3.30278 by substitiping it back into the initial equation.

Step 8: Verify Root

By substituting \(x=3.30278\) back into the original equation, it's shown to be correct. So there is only one solution for the given equation which is \(x=3.30278\)

Key Concepts

Properties of Logarithms

When working with logarithmic expressions, understanding their properties can significantly simplify your calculations. One of the most fundamental properties is the subtraction of logarithms, expressed as \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). This equation shows that when you subtract one logarithm from another, it is equivalent to the logarithm of the division of their arguments. In the original exercise, this property helps convert the more complex expression \( \ln (x+1) - \ln (x-2) \) into a single logarithm \( \ln \left(\frac{x+1}{x-2}\right) \).

• The quotient rule of logarithms can help reduce and simplify expressions.
• Making calculations simpler and easier to handle.

Furthermore, by expressing both sides of the equation as a single logarithm, the problem becomes more straightforward to solve because it removes the logarithms altogether when equating the internal expressions.

Quadratic Formula

After simplifying the logarithmic equation, you may find yourself left with a quadratic equation like \( (x+1)=x(x-2) \). To solve this, you can rearrange it to a standard form as \( x^2 - 3x - 1 = 0 \). Quadratic equations of this type can be solved using the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \( a \), \( b \), and \( c \) represent the coefficients from the quadratic equation.

• This formula is especially useful for equations that cannot be factored easily.

With the given coefficients \( a = 1 \), \( b = -3 \), and \( c = -1 \), substitute them into the formula to find the solutions. This process will yield roots, which are crucial to determine the values of \( x \) that satisfy the equation.

Domain of Logarithmic Functions

The domain of a logarithmic function is crucial to ensure the validity of your solutions. Logarithms are defined only for positive numbers, meaning the argument inside a logarithm must be greater than zero for it to exist. This constraint creates particular attention when solving equations involving logarithms.

For example, our solution gives us two roots: \( x = 3.30278 \) and \( x = -0.30278 \). Here, \( x = -0.30278 \) must be rejected since substituting it into the original logarithmic terms would result in taking the log of a negative number, which is undefined.

• Always verify if the obtained roots fall within the domain of the original logarithmic equation.
• This ensures that the solution is not just mathematically correct, but also meaningful.

Checking the domain helps exclude extraneous solutions that may arise from algebraic manipulations, securing only the root that is valid within the problem’s constraints.