Problem 54 - Elementary and Intermediate Algebra

Question

Solve each equation. See Example \(8 .\) $$ \log x+\log (x+9)=1 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 1 \).

Step 1: Apply Logarithmic Property

The equation given is \( \log x + \log (x+9) = 1 \). Use the property of logarithms that states \( \log a + \log b = \log(ab) \). This allows us to combine the logs: \( \log(x(x+9)) = 1 \).

Step 2: Remove the Logarithm

Rewrite the equation in exponential form to eliminate the logarithm. Since \( \log_{10} a = b \) means \( a = 10^b \), we have \( x(x+9) = 10^1 \), which simplifies to \( x(x+9) = 10 \).

Step 3: Expand and Rearrange the Equation

Expand the left side of the equation: \( x^2 + 9x = 10 \). Rearrange the equation to form a standard quadratic equation: \( x^2 + 9x - 10 = 0 \).

Step 4: Solve the Quadratic Equation

To solve \( x^2 + 9x - 10 = 0 \), we can use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 9 \), and \( c = -10 \). Calculate the discriminant: \( b^2 - 4ac = 9^2 - 4(1)(-10) = 81 + 40 = 121 \).

Step 5: Calculate the Roots

The discriminant is a perfect square (121), so the roots are real and rational. Substitute into the quadratic formula: \( x = \frac{-9 \pm \sqrt{121}}{2} \), which simplifies to \( x = \frac{-9 \pm 11}{2} \).

Step 6: Determine the Solutions

Compute the two possible \( x \) values: \( x = \frac{-9 + 11}{2} = 1 \) and \( x = \frac{-9 - 11}{2} = -10 \). However, since \( x \) must be positive for the logarithm to be defined, the only valid solution is \( x = 1 \).

Key Concepts

Quadratic Equations

Quadratic equations come up when solving various algebraic problems, including those involving logarithms. A quadratic equation is an algebraic equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. The term "quadratic" comes from "quad" signifying "squared," as the highest power of \( x \) is 2.
**Solving Quadratic Equations:**
When you encounter a quadratic equation, there are several methods you can use to find the solution(s):

• Factoring: If the quadratic can be factored, you can set each factor equal to zero and solve for \( x \).
• Quadratic Formula: The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides the roots of the quadratic, even when factoring is difficult or impossible.
• Completing the Square: This method involves rewriting the equation so that one side is a perfect square trinomial.

In the given logarithmic problem, after converting the logarithmic equation to a quadratic form \( x^2 + 9x - 10 = 0 \), we used the quadratic formula because it always works, especially when the expression is not easily factorable.

The use of the quadratic formula allowed us to find the roots: \( x = 1 \) and \( x = -10 \). However, in the context of logarithms, only positive solutions are valid due to the domain of logarithmic functions.

Properties of Logarithms

Understanding the properties of logarithms can simplify complex problems and reveal the underlying structure of equations. Logarithmic properties are powerful tools that transform multi-step problems into simpler forms.
**Main Properties of Logarithms Used:**
In the provided exercise, the property \( \log a + \log b = \log(ab) \), also known as the "product property," is employed.

• This property states that the log of a product is the sum of the logs. It helps to combine multiple logarithms into a single expression.
• In the exercise, \( \log x + \log(x+9) = 1 \) converts to \( \log(x(x+9)) = 1 \), simplifying the equation significantly.

To solve logarithmic equations effectively, knowing when to apply these properties is crucial. For beginners, it helps to remember:

• Product Property: \( \log_b(m \, n) = \log_b m + \log_b n \)
• Quotient Property: \( \log_b\left( \frac{m}{n} \right) = \log_b m - \log_b n \)
• Power Property: \( \log_b(m^n) = n \log_b m \)

Using these properties ensures that you can simplify equations and make them more manageable for further manipulation.

Exponential Form

Exponential form is a key concept when connecting logarithms and equations. It is the inverse of taking a logarithm, and useful for solving logarithmic equations.
**Transforming to Exponential Form:**
The basic idea is to rewrite a logarithmic equation \( \log_b a = c \) as its exponential form \( a = b^c \). This conversion is crucial in revealing solutions:

• Within the exercise, once the logs were combined, we rewrote \( \log(x(x+9)) = 1 \) as \( x(x+9) = 10^1 \) or simply \( x(x+9) = 10 \).
• This transformation allowed us to eliminate the logarithm and work directly with the equation, making it easier to manipulate into a quadratic equation form.

Understanding how and when to use exponential transformations is essential. This technique is often used to "remove" the logarithm so that traditional algebraic methods, such as factoring or using the quadratic formula, can be applied.
By understanding exponential forms and their relationship with logarithms, you gain greater flexibility in manipulating and solving equations that initially seem complex or daunting.