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Magazine Chapter 9: Problem 39
Solve each equation. See Examples 1 through 4. $$ \log _{2} x+\log _{2}(x+5)=1 $$
Short Answer
Expert verified
\( x \approx 0.372 \)
Step by step solution
01
Apply the Product Rule for Logarithms
The problem involves a sum of logarithms with the same base, which allows us to apply the product rule for logarithms: \( \log_b(m) + \log_b(n) = \log_b(mn) \). Here, apply this rule to rewrite the equation as a single logarithm: \( \log_2(x(x+5)) = 1 \).
02
Convert the Logarithmic Equation to Exponential Form
Convert the logarithmic equation \( \log_2(x(x+5)) = 1 \) into its exponential form. The equation now becomes \( x(x+5) = 2^1 \), which simplifies to \( x(x+5) = 2 \).
03
Simplify and Form a Quadratic Equation
Distribute and simplify the equation \( x(x+5) = 2 \) to form a quadratic equation: \( x^2 + 5x = 2 \). Rearrange it to \( x^2 + 5x - 2 = 0 \).
04
Solve the Quadratic Equation Using the Quadratic Formula
Employ the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1 \), \( b = 5 \), and \( c = -2 \) to solve \( x^2 + 5x - 2 = 0 \). This gives \( x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-2)}}{2(1)} = \frac{-5 \pm \sqrt{33}}{2} \).
05
Evaluate the Solutions and Check for Validity
Evaluate \( \frac{-5 \pm \sqrt{33}}{2} \) to find approximate numerical values: \( x \approx 0.372 \) and \( x \approx -5.372 \). Since logarithms of negative numbers are undefined in real numbers, check the solutions in the context of the logarithmic function. Only \( x \approx 0.372 \) is valid because \( x + 5 > 0 \) for \( x \approx 0.372 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
A quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \). Here are a few things to remember about these equations:
- The term \( ax^2 \) is the quadratic term because it contains the variable squared.
- The linear term is \( bx \), and \( c \) is the constant term.
- Solving the quadratic equation means finding the values for \( x \) where the equation equals zero, often referred to as the 'roots' of the equation.
Product Rule for Logarithms
The product rule for logarithms is a useful property when you encounter a sum of logarithms with the same base. It states that \( \log_b(m) + \log_b(n) = \log_b(mn) \). This means you can combine the logarithms into one by multiplying their arguments.
- This rule simplifies complex logarithmic equations by reducing multiple logarithms into a single statement.
- In our original exercise, \( \log_2 x + \log_2(x+5) \) is transformed into \( \log_2(x(x+5)) \) using the product rule, greatly simplifying subsequent steps.
Exponential Form
Converting logarithmic equations to exponential form is a critical step in solving them. The basic idea revolves around understanding the relationship between logarithms and exponents. The change from logarithmic to exponential form is given by \( \log_b(a) = c \) becomes \( b^c = a \).
- This transformation shifts the focus from logarithmic equations to exponential ones, which are often simpler to solve.
- In the solution provided, \( \log_2(x(x+5)) = 1 \) becomes \( 2^1 = x(x+5) \), rewriting it into \( x(x+5) = 2 \).
Quadratic Formula
The quadratic formula is a powerful tool for finding the roots of any quadratic equation. Given a standard form \( ax^2 + bx + c = 0 \), the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides the solution for \( x \).
- It is derived from the process of completing the square and is universally applicable to all quadratic equations.
- In the discussed exercise, the quadratic formula helps solve \( x^2 + 5x - 2 = 0 \).
- By substituting \( a = 1 \), \( b = 5 \), and \( c = -2 \), the formula calculates the potential roots: \( x = \frac{-5 \pm \sqrt{33}}{2} \).
- Since logarithms only accept positive numbers, the validity of these roots must be checked accordingly.
