Solving Logarithmic Equations A Step-by-Step Guide To Log(x) + Log(x+2) = 1
This comprehensive guide delves into the intricacies of solving logarithmic equations, focusing specifically on the equation log(x) + log(x+2) = 1. We will explore the fundamental principles governing logarithms, the properties that enable us to manipulate and simplify logarithmic expressions, and the step-by-step process of solving the given equation. This article is designed for students, educators, and anyone seeking a deeper understanding of logarithmic functions and their applications. Logarithmic equations might seem daunting at first, but by understanding the underlying principles and applying them systematically, we can conquer any logarithmic challenge.
Understanding Logarithms
To effectively solve logarithmic equations, we must first establish a solid understanding of what logarithms are and how they function. At its core, a logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise the base to obtain a certain number?" In mathematical terms, if we have the equation by = x, then the logarithm of x to the base b is written as logb(x) = y. Here, b is the base, x is the argument (the number we're taking the logarithm of), and y is the exponent.
For instance, log10(100) = 2 because 10 raised to the power of 2 equals 100. Similarly, log2(8) = 3 because 2 raised to the power of 3 equals 8. When the base is not explicitly written, as in log(x), it is generally understood to be base 10, also known as the common logarithm. Another important base is e (Euler's number, approximately 2.71828), which gives us the natural logarithm, denoted as ln(x).
Logarithms possess several key properties that are instrumental in simplifying and solving logarithmic equations. These properties stem directly from the properties of exponents. Let's explore some of the most crucial ones:
- Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, logb(mn) = logb(m) + logb(n).
- Quotient Rule: The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. Mathematically, logb(m/n) = logb(m) - logb(n).
- Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, logb(mp) = p logb(m).
- Change of Base Formula: This rule allows us to convert logarithms from one base to another. Mathematically, loga(x) = logb(x) / logb(a).
These properties are the building blocks for manipulating logarithmic expressions and solving logarithmic equations. Mastering them is essential for anyone seeking to tackle more complex problems involving logarithms. In the context of solving the equation log(x) + log(x+2) = 1, we will primarily utilize the product rule and the fundamental relationship between logarithms and exponents.
Solving log(x) + log(x+2) = 1: A Step-by-Step Approach
Now, let's apply our understanding of logarithms to solve the equation log(x) + log(x+2) = 1. This equation involves the sum of two logarithms, which we can simplify using the product rule. Remember, when the base is not explicitly written, it is understood to be 10. Therefore, we are working with common logarithms.
Step 1: Apply the Product Rule
The product rule states that logb(mn) = logb(m) + logb(n). Applying this rule to our equation, we can combine the two logarithmic terms into a single logarithm:
log(x) + log(x+2) = log(x(x+2)).
So, our equation now becomes:
log(x(x+2)) = 1.
This step significantly simplifies the equation by reducing two logarithmic terms into one, making it easier to manipulate.
Step 2: Convert to Exponential Form
To eliminate the logarithm, we need to convert the equation from logarithmic form to exponential form. Recall the fundamental relationship between logarithms and exponents: logb(x) = y is equivalent to by = x. In our case, the base is 10, the argument is x(x+2), and the exponent is 1. Therefore, we can rewrite the equation as:
101 = x(x+2).
This step effectively removes the logarithm, transforming the equation into a more familiar algebraic form.
Step 3: Simplify and Rearrange into a Quadratic Equation
Now we have a simple exponential equation that we can easily solve. First, simplify the left side:
10 = x(x+2).
Next, expand the right side:
10 = x2 + 2x.
To solve for x, we need to rearrange the equation into a quadratic equation in the standard form ax2 + bx + c = 0. Subtract 10 from both sides:
0 = x2 + 2x - 10.
We now have a quadratic equation that we can solve using various methods, such as factoring, completing the square, or the quadratic formula.
Step 4: Solve the Quadratic Equation
The quadratic equation we obtained is x2 + 2x - 10 = 0. This equation does not factor easily, so we will use the quadratic formula to find the solutions. The quadratic formula is given by:
x = (-b ± √(b2 - 4ac)) / 2a.
In our equation, a = 1, b = 2, and c = -10. Substituting these values into the quadratic formula, we get:
x = (-2 ± √(22 - 4(1)(-10))) / 2(1).
Simplify the expression under the square root:
x = (-2 ± √(4 + 40)) / 2.
x = (-2 ± √44) / 2.
Further simplify the square root:
x = (-2 ± 2√11) / 2.
Finally, divide both terms in the numerator by 2:
x = -1 ± √11.
This gives us two potential solutions: x = -1 + √11 and x = -1 - √11.
Step 5: Check for Extraneous Solutions
It is crucial to check our solutions in the original equation to ensure they are valid. Logarithmic functions are only defined for positive arguments. This means that the values inside the logarithms must be greater than zero. In our original equation, log(x) + log(x+2) = 1, we have two logarithmic terms: log(x) and log(x+2). Therefore, both x and x+2 must be greater than zero.
Let's analyze our potential solutions:
- x = -1 + √11: Since √11 is approximately 3.317, x is approximately -1 + 3.317 = 2.317, which is greater than zero. Also, x+2 would be approximately 4.317, which is also greater than zero. So, this solution is potentially valid.
- x = -1 - √11: This value is approximately -1 - 3.317 = -4.317, which is less than zero. Since we cannot take the logarithm of a negative number, this solution is extraneous and must be discarded.
Therefore, the only valid solution is x = -1 + √11.
Conclusion
In this detailed guide, we have successfully solved the logarithmic equation log(x) + log(x+2) = 1. We began by understanding the fundamental principles of logarithms and their properties, particularly the product rule. We then applied the product rule to simplify the equation, converted it to exponential form, and rearranged it into a quadratic equation. Using the quadratic formula, we found two potential solutions, but after checking for extraneous solutions, we determined that the only valid solution is x = -1 + √11. This process highlights the importance of understanding logarithmic properties and the need to verify solutions in the context of the original equation.
Solving logarithmic equations requires a systematic approach and a solid understanding of the underlying principles. By mastering these concepts, you can confidently tackle a wide range of logarithmic problems and appreciate the power and versatility of logarithmic functions in mathematics and various real-world applications. This exploration of log(x) + log(x+2) = 1 serves as a valuable example of the techniques and considerations involved in solving logarithmic equations, providing a foundation for further exploration and mastery of this important mathematical concept.
