Problem 60 - College Algebra with Corequisite Support

Question

Use properties of exponents to find the \(x\) -intercepts of the function \(f(x)=\log \left(x^{2}+4 x+4\right)\) algebraically. Show the steps for solving, and then verify the result by graphing the function.

Step-by-Step Solution

Answer

The x-intercepts are at x = -3 and x = -1.

Step 1: Understanding the problem

We are given the function \(f(x)=\log(x^2+4x+4)\). The goal is to find the \(x\)-intercepts, which are the points where the function equals zero. These intercepts occur when the expression inside the logarithm equals one, because \(\log(1) = 0\).

Step 2: Setting up the equation

To find the \(x\)-intercepts of the logarithmic function, we set the expression inside the logarithm equal to one: \(x^2 + 4x + 4 = 1\).

Step 3: Simplifying the equation

Rearrange and simplify the equation by subtracting one from both sides: \(x^2 + 4x + 3 = 0\).

Step 4: Factoring the quadratic

Factor the quadratic equation \(x^2 + 4x + 3 = 0\). This factors to \((x+3)(x+1) = 0\).

Step 5: Finding the solutions

Set each factor equal to zero to solve for \(x\): \(x+3 = 0\) and \(x+1 = 0\). Solving these gives \(x = -3\) and \(x = -1\).

Step 6: Verification by graphing

Graph the function \(f(x) = \log(x^2+4x+4)\) to verify the intercepts. The graph should intersect the x-axis at \(x = -3\) and \(x = -1\). Ensure the graph visually confirms the roots found algebraically.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions, and they play a crucial role in algebra and calculus. A logarithm tells us the power to which a number, called the base, must be raised to produce a given number. For instance, in the expression \( \log_b(y) = x \), \( b^x = y \). This means that \( x \) is the power to which the base \( b \) must be raised to produce \( y \).

Some key properties of logarithmic functions include:

• The logarithm of 1 (with any base) is 0: \( \log_b(1) = 0 \), because any non-zero number to the power of 0 is 1.
• The logarithm of the base itself is 1: \( \log_b(b) = 1 \), because \( b^1 = b \).
• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \).
• Quotient Property: \( \log_b\left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \).
• Power Property: \( \log_b(M^p) = p \log_b(M) \).

These properties are essential when solving logarithmic equations, like converting the problem \( \log(x^2+4x+4)=0 \) into a simpler form where the expression equals to 1.

Quadratic Equations

A quadratic equation is a type of polynomial equation with the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations are fundamental in algebra, allowing us to easily find unknown values by finding the roots of the equation.

There are several methods to solve a quadratic equation, including:

• Factoring: This involves expressing the quadratic equation as a product of two binomials.
• Quadratic Formula: This formula, \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \), is a universal method for solving any quadratic equation.
• Completing the Square: This method involves rewriting a quadratic in the form \( (x-h)^2 = k \).

Using these approaches, especially factoring, can simplify finding solutions. For instance, solving \( x^2 + 4x + 3 = 0 \) by factoring helps to quickly find the x-intercepts needed in solving logarithmic function problems.

Factoring Quadratics

Factoring quadratics is a method used to break down equations of the form \( ax^2 + bx + c = 0 \) into a product of binomial expressions. This technique is particularly useful because once factored, you can solve for the variable by applying the zero-product property.

In the example \( x^2 + 4x + 3 \), we look to factor the equation into the form \( (x + m)(x + n) = 0 \). The numbers \( m \) and \( n \) should multiply to give \( c \) (the constant term) and add up to give \( b \) (the coefficient of the linear term). In this case, \( m = 3 \) and \( n = 1 \).

Therefore, the equation factors to \( (x + 3)(x + 1) = 0 \).

Once factored, set each factor equal to zero: \( x + 3 = 0 \) and \( x + 1 = 0 \), giving solutions \( x = -3 \) and \( x = -1 \). These solutions provide the x-intercepts of the original function \( f(x) = \log(x^2+4x+4) \).

Mastering factoring helps quickly and efficiently solve many algebraic problems, especially in contexts involving quadratic equations and their applications.