Solving Logarithmic Equations Log₂(−9r − 22) = Log₂(−5r − 10) A Step-by-Step Guide
In the realm of mathematics, solving logarithmic equations is a fundamental skill. Logarithmic equations appear in various contexts, from calculus to physics, making it essential to understand how to approach and solve them effectively. This article delves into the step-by-step solution of the logarithmic equation log₂(−9r − 22) = log₂(−5r − 10), providing a comprehensive understanding of the underlying principles and techniques involved.
Understanding Logarithmic Equations
To successfully solve the equation, it's crucial to grasp the nature of logarithmic equations. A logarithmic equation is an equation that contains logarithms. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. In simpler terms, if we have logₐ(b) = c, it means a raised to the power of c equals b (aᶜ = b). Understanding this relationship is the cornerstone of solving logarithmic equations.
Key Properties of Logarithms
Before diving into the solution, it's beneficial to review some key properties of logarithms that will be used:
- Logarithmic Equality: If logₐ(x) = logₐ(y), then x = y, provided a > 0 and a ≠ 1.
- Logarithmic Definition: The logarithm logₐ(x) is defined only when x > 0 and a > 0, a ≠ 1.
These properties provide the foundation for simplifying and solving logarithmic equations. Keeping these in mind ensures the solutions obtained are valid and meaningful.
Step-by-Step Solution of log₂(−9r − 22) = log₂(−5r − 10)
Now, let's solve the equation log₂(−9r − 22) = log₂(−5r − 10) step by step.
Step 1: Apply the Logarithmic Equality Property
Since the bases of the logarithms are the same (base 2), we can apply the logarithmic equality property. This property states that if logₐ(x) = logₐ(y), then x = y. Applying this to our equation, we get:
−9r − 22 = −5r − 10
This step simplifies the equation by eliminating the logarithms, making it easier to solve for the variable 'r'.
Step 2: Rearrange the Equation
Next, we rearrange the equation to group the terms involving 'r' on one side and the constant terms on the other. This is a standard algebraic technique to isolate the variable. To do this, we can add 9r to both sides and add 10 to both sides of the equation:
−9r − 22 + 9r + 10 = −5r − 10 + 9r + 10
This simplifies to:
−12 = 4r
By grouping like terms, we've brought the equation closer to a form where 'r' can be easily determined.
Step 3: Solve for 'r'
To isolate 'r', we divide both sides of the equation by 4:
−12 / 4 = 4r / 4
This gives us:
r = -3
At this stage, we have found a potential solution for 'r'. However, it's crucial to verify this solution to ensure it is valid within the domain of the logarithmic equation.
Step 4: Check the Solution
The final and critical step is to check the solution in the original logarithmic equation. Remember, logarithms are only defined for positive arguments. We need to ensure that substituting r = -3 into the original equation does not result in taking the logarithm of a negative number or zero.
Original equation:
log₂(−9r − 22) = log₂(−5r − 10)
Substitute r = -3:
log₂(−9(-3) − 22) = log₂(−5(-3) − 10)
Simplify:
log₂(27 − 22) = log₂(15 − 10)
log₂(5) = log₂(5)
Since both arguments of the logarithms are positive (5 > 0), the solution r = -3 is valid.
Conclusion
In summary, the solution to the logarithmic equation log₂(−9r − 22) = log₂(−5r − 10) is r = -3. This solution was obtained by applying the logarithmic equality property, rearranging the equation, solving for 'r', and crucially, verifying the solution in the original equation to ensure its validity. Understanding these steps and the properties of logarithms is key to solving similar logarithmic equations effectively. Solving logarithmic equations is not just about finding a numerical answer; it's about understanding the mathematical principles and constraints that govern these equations. By mastering these concepts, one can tackle more complex mathematical problems with confidence.
Logarithmic equations, often perceived as challenging, are a fundamental part of mathematics with applications across various fields. The equation log₂(−9r − 22) = log₂(−5r − 10) presents an excellent opportunity to delve into the intricacies of logarithmic functions and their solutions. This comprehensive guide aims to break down the process, ensuring a clear understanding of the methods involved and the underlying principles of logarithmic equations.
Unveiling the Essence of Logarithms
At its core, solving logarithmic equations requires a solid grasp of what logarithms represent. A logarithm is essentially the inverse operation of exponentiation. The expression logₐ(b) = c means that 'a' raised to the power of 'c' equals 'b'. In mathematical notation, this is expressed as aᶜ = b. Understanding this relationship is crucial as it forms the basis for manipulating and solving logarithmic equations. The base 'a' plays a critical role, and it must always be a positive number not equal to 1.
Fundamental Logarithmic Properties
To effectively solve logarithmic equations, one must be familiar with several fundamental logarithmic properties. These properties act as tools that simplify complex equations and pave the way for solutions. Here are some key properties:
- Product Rule: logₐ(mn) = logₐ(m) + logₐ(n)
- Quotient Rule: logₐ(m/n) = logₐ(m) − logₐ(n)
- Power Rule: logₐ(mᵖ) = p * logₐ(m)
- Change of Base Formula: log_b(a) = log_c(a) / log_c(b)
- Logarithmic Equality: If logₐ(x) = logₐ(y), then x = y, provided a > 0 and a ≠ 1.
These properties are not just theoretical concepts; they are practical tools that aid in transforming and simplifying logarithmic expressions. Recognizing when and how to apply these properties is a key skill in solving logarithmic equations.
Step-by-Step Solution of log₂(−9r − 22) = log₂(−5r − 10): A Detailed Approach
Let’s tackle the equation log₂(−9r − 22) = log₂(−5r − 10) with a step-by-step approach, ensuring clarity and precision at each stage.
Step 1: Utilizing the Logarithmic Equality Property
The equation presents us with a scenario where the logarithms on both sides have the same base (base 2). This allows us to directly apply the logarithmic equality property. As mentioned earlier, if logₐ(x) = logₐ(y), then x = y, provided a > 0 and a ≠ 1. Applying this property, we can eliminate the logarithms and set the arguments equal to each other:
−9r − 22 = −5r − 10
This simplification transforms the logarithmic equation into a linear equation, making it more manageable to solve.
Step 2: Isolating the Variable 'r'
Our next goal is to isolate the variable 'r'. To do this, we need to gather all terms involving 'r' on one side of the equation and the constants on the other side. This is a standard algebraic technique. We can achieve this by adding 9r to both sides and adding 10 to both sides of the equation:
−9r − 22 + 9r + 10 = −5r − 10 + 9r + 10
Simplifying this, we get:
−12 = 4r
The equation is now in a simpler form, with 'r' appearing only on one side. This brings us closer to finding the value of 'r'.
Step 3: Solving for 'r' Algebraically
To find the value of 'r', we need to isolate 'r' completely. Currently, 'r' is multiplied by 4. To undo this multiplication, we divide both sides of the equation by 4:
−12 / 4 = 4r / 4
Performing the division gives us:
r = -3
At this point, we have found a potential solution for 'r'. However, it's essential to remember that finding a solution is not the end of the process. We must verify this solution to ensure it is valid within the original equation's domain.
Step 4: Verification: The Key to Valid Solutions
Verifying the solution is a critical step in solving logarithmic equations. Logarithms have domain restrictions: the argument of a logarithm must be positive. Substituting r = -3 back into the original equation, we must ensure that the arguments of the logarithms remain positive.
Let’s substitute r = -3 into the original equation:
log₂(−9r − 22) = log₂(−5r − 10)
log₂(−9(-3) − 22) = log₂(−5(-3) − 10)
Simplifying the expressions inside the logarithms:
log₂(27 − 22) = log₂(15 − 10)
log₂(5) = log₂(5)
The arguments of the logarithms are both 5, which is a positive number. Therefore, r = -3 is a valid solution.
Final Thoughts: The Art of Solving Logarithmic Equations
The solution to the equation log₂(−9r − 22) = log₂(−5r − 10) is r = -3. This was achieved by applying the logarithmic equality property, simplifying the equation, solving for 'r', and rigorously verifying the solution. Solving logarithmic equations is an art that blends algebraic manipulation with a deep understanding of logarithmic properties and domain restrictions. By mastering these elements, one can confidently navigate the world of logarithmic equations and their diverse applications in mathematics and beyond.
Solving logarithmic equations might appear daunting initially, but with a systematic approach, they can be demystified. This article provides a detailed walkthrough on solving the equation log₂(−9r − 22) = log₂(−5r − 10), emphasizing the underlying mathematical principles and techniques.
Grasping the Fundamentals of Logarithms
Before diving into the solution, it's crucial to grasp the fundamentals of logarithms. A logarithm is the inverse operation to exponentiation. The logarithm of a number to a given base is the exponent to which we must raise the base to produce that number. For instance, logₐ(b) = c implies that a raised to the power of c equals b, written as aᶜ = b. This understanding is foundational for manipulating and solving logarithmic equations.
Essential Logarithmic Properties and Rules
To effectively solve logarithmic equations, familiarity with essential logarithmic properties and rules is paramount. These properties serve as tools to simplify equations and reveal their solutions. Some of the key properties include:
- Product Rule: logₐ(xy) = logₐ(x) + logₐ(y)
- Quotient Rule: logₐ(x/y) = logₐ(x) − logₐ(y)
- Power Rule: logₐ(xⁿ) = n * logₐ(x)
- Change of Base Formula: log_b(a) = log_c(a) / log_c(b)
- Equality Rule: If logₐ(x) = logₐ(y), then x = y (provided a > 0 and a ≠ 1)
These properties are not just theoretical constructs; they are practical tools for simplifying logarithmic expressions. Recognizing when and how to apply these properties is a critical skill in solving logarithmic equations.
Solving log₂(−9r − 22) = log₂(−5r − 10): A Step-by-Step Guide
Let's proceed with a step-by-step solution of the equation log₂(−9r − 22) = log₂(−5r − 10).
Step 1: Applying the Equality Rule of Logarithms
The equation presents a scenario where the logarithms on both sides have the same base (2). This allows us to apply the equality rule of logarithms. If logₐ(x) = logₐ(y), then x = y (for a > 0 and a ≠ 1). Applying this rule to our equation, we equate the arguments:
−9r − 22 = −5r − 10
This step simplifies the equation significantly, transforming it from a logarithmic equation into a linear equation.
Step 2: Rearranging the Linear Equation
Next, we need to rearrange the linear equation to isolate the variable 'r'. This involves grouping like terms together. We can add 9r to both sides and add 10 to both sides:
−9r − 22 + 9r + 10 = −5r − 10 + 9r + 10
Simplifying, we get:
−12 = 4r
Now, the equation is in a more manageable form, with terms involving 'r' on one side and constants on the other.
Step 3: Solving for the Variable 'r'
To find the value of 'r', we need to isolate 'r' completely. Currently, 'r' is multiplied by 4. To isolate 'r', we divide both sides of the equation by 4:
−12 / 4 = 4r / 4
This gives us:
r = -3
We have found a potential solution for 'r'. However, before declaring this as the final answer, it is crucial to verify this solution in the original equation.
Step 4: Verifying the Solution's Validity
Verifying the solution is a critical step in solving logarithmic equations. Logarithms are only defined for positive arguments. Thus, we must ensure that substituting r = -3 into the original equation does not result in taking the logarithm of a non-positive number.
Substituting r = -3 into the original equation:
log₂(−9r − 22) = log₂(−5r − 10)
log₂(−9(-3) − 22) = log₂(−5(-3) − 10)
Simplifying the expressions inside the logarithms:
log₂(27 − 22) = log₂(15 − 10)
log₂(5) = log₂(5)
Both arguments of the logarithms are 5, which is a positive number. Hence, r = -3 is a valid solution.
Conclusion: Mastering the Art of Solving Logarithmic Equations
The solution to the equation log₂(−9r − 22) = log₂(−5r − 10) is r = -3. This solution was obtained by applying the equality rule of logarithms, rearranging the equation, solving for 'r', and, importantly, verifying the solution's validity. Solving logarithmic equations is not just about algebraic manipulation; it's about understanding the properties and constraints of logarithms. With a clear understanding of these principles and a systematic approach, one can confidently solve a wide range of logarithmic equations.
