Problem 53 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(8 .\) $$ \log x+\log (x-48)=2 $$

Step-by-Step Solution

Answer

The solution is \(x = 50\).

Step 1: Recognize the Logarithm Property

We begin by recognizing the property of logarithms that states \(\log a + \log b = \log(ab)\). We can use this property to combine the logarithms on the left-hand side of the equation.

Step 2: Apply the Logarithm Property

Apply the property to combine the logarithms: \(\log x + \log(x-48) = \log(x(x-48))\).The equation becomes \(\log(x^2 - 48x) = 2\).

Step 3: Convert to Exponential Form

Convert the logarithmic equation \(\log(x^2 - 48x) = 2\) into exponential form. The equation \(\log_b(a) = c\) can be rewritten as \(a = b^c\). Here, our base \(b\) is 10, so: \(x^2 - 48x = 10^2\).

Step 4: Simplify the Equation

Simplify and solve the resulting equation: \(x^2 - 48x = 100\).Rearrange it to form a standard quadratic equation: \(x^2 - 48x - 100 = 0\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula to solve the equation \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -48\), \(c = -100\). Calculate the discriminant: \(\Delta = b^2 - 4ac = (-48)^2 - 4 \times 1 \times (-100) = 2304 + 400 = 2704\). Find the square root and complete solving for \(x\).

Step 6: Calculate the Roots

Calculate the roots using the quadratic formula: \(x = \frac{-(-48) \pm \sqrt{2704}}{2}\).Find the square root: \(\sqrt{2704} = 52\). Thus, the solutions are \(x = \frac{48 \pm 52}{2}\).

Step 7: Check the Solutions

Calculate: \(x = \frac{48 + 52}{2} = 50\) and \(x = \frac{48 - 52}{2} = -2\).Since a logarithm's argument must be positive, discard \(x = -2\) since \(\log(-2)\) is undefined. Hence, the valid solution is \(x = 50\).

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms of a variable or expression. The main challenge in solving logarithmic equations often lies in removing the logarithms so the variable can be isolated. This is commonly achieved using the properties of logarithms. One such property is

• Addition property: If you have multiple logarithms added together, like \( \log a + \log b \), you can combine them into one logarithm: \( \log(ab) \).

Make sure the arguments (the numbers or expressions inside the logarithm) remain positive, as the logarithm of a negative number is undefined. Recognizing when and how to use these properties is crucial for simplifying the equation to a form that can be more easily manipulated.

Exponential Form Conversion

Once the logarithms are combined, the equation is often converted into exponential form. This is done because exponential equations are typically easier to solve. You may recall that a logarithmic equation of the form \( \log_b(a) = c \) can be rewritten as \( a = b^c \).

• The base \( b \) is the number that the logarithm is using.
• The argument \( a \) becomes the result of raising the base to the power of \( c \).

In this exercise, the base is assumed to be 10 because it is a common logarithm format if not otherwise specified. Therefore, the equation \( \log(x^2 - 48x) = 2 \) is converted to \( x^2 - 48x = 10^2 \).
This step eliminates the logarithm, simplifying the equation and making it possible to isolate the variable.

Discriminant Calculation

Before using the quadratic formula, it's important to calculate the discriminant to determine the nature of the roots of the quadratic equation. The discriminant, denoted as \( \Delta \), is found using the formula:

• \( \Delta = b^2 - 4ac \)

where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The discriminant has critical implications:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (a repeated root).
• If \( \Delta < 0 \), the roots are not real numbers but rather complex.

Here, we calculated \( \Delta = 2704 \), indicating two distinct real solutions.

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. Once you have calculated the discriminant, you can use the formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula yields the potential values of \( x \) which solve the equation. In our example, the coefficients are \( a = 1 \), \( b = -48 \), and \( c = -100 \).
We substitute these into the quadratic formula, solving for \( x \). It's important to calculate \( \sqrt{\Delta} \), in this case \( \sqrt{2704} = 52 \), to apply it correctly.
The results of the quadratic formula will give you potential solutions, which should be tested to ensure they fit within the domain of the original equation, especially considering the constraints of logarithms.