Solve Log₂(x² + 5) = 3: A Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of logarithmic equations. We're going to break down how to solve the equation log₂(x² + 5) = 3 step-by-step. This kind of problem might seem a bit tricky at first, but don't worry, we'll make it super clear and straightforward. By the end of this article, you'll be a pro at solving these types of equations. So, grab your thinking caps, and let's get started!
Understanding Logarithmic Equations
Before we jump into solving our specific equation, let's make sure we're all on the same page about what logarithmic equations actually are. In simple terms, a logarithmic equation is an equation that involves a logarithm. Logarithms are essentially the inverse operation to exponentiation. Think of it like this: if 2³ = 8, then log₂8 = 3. The logarithm tells you what exponent you need to raise the base (in this case, 2) to, in order to get a certain number (in this case, 8).
Why are logarithms important? Well, they pop up in all sorts of places, from calculating the magnitude of earthquakes (the Richter scale) to understanding the pH levels of chemicals. They're also crucial in many areas of mathematics, science, and engineering. So, mastering logarithms is definitely a valuable skill. When it comes to the logarithmic equation, it's essential to understand the relationship between logarithms and exponents. The equation logₐb = c is equivalent to aᶜ = b. This conversion is the key to solving many logarithmic problems. Understanding the relationship between exponents and logarithms will not only help you solve equations but also give you a deeper insight into mathematical concepts. By grasping this fundamental principle, you can approach logarithmic problems with confidence and ease.
Different bases for logarithms exist, such as the common logarithm (base 10) and the natural logarithm (base e), each with its own applications and properties. Recognizing these differences and knowing when to use each type is crucial for accurate problem-solving. Logarithms are used in various fields, including computer science, where they help analyze the efficiency of algorithms, and finance, where they are used in calculations involving compound interest. So, learning about logarithms is not just about solving equations; it's about gaining a versatile tool that will be useful in many different contexts.
Step-by-Step Solution for log₂(x² + 5) = 3
Okay, let's tackle our equation: log₂(x² + 5) = 3. Here's a step-by-step breakdown of how to solve it:
Step 1: Convert the Logarithmic Equation to Exponential Form
Remember that logₐb = c is the same as aᶜ = b. So, we can rewrite our equation log₂(x² + 5) = 3 in exponential form. Our base is 2, the exponent is 3, and the result is x² + 5. This gives us:
2³ = x² + 5
This transformation is the cornerstone of solving logarithmic equations. By converting the equation into exponential form, we eliminate the logarithm and create a more manageable algebraic expression. It's like translating from one language to another – once we have the equation in a familiar form, the rest of the solution becomes much clearer. This step highlights the inverse relationship between logarithms and exponents, a relationship that is fundamental to understanding and working with logarithmic functions. By mastering this conversion, you'll be well-equipped to tackle a wide variety of logarithmic equations.
Step 2: Simplify the Exponential Expression
Now, let's simplify 2³. We know that 2³ = 2 * 2 * 2 = 8. So our equation becomes:
8 = x² + 5
Simplifying the exponential expression is a crucial step as it reduces the complexity of the equation, making it easier to solve for the variable. In this case, calculating 2³ as 8 transforms the equation into a more recognizable quadratic form. This step not only simplifies the arithmetic but also sets the stage for applying algebraic techniques to isolate the variable. Accuracy in simplification is key, as any error here can propagate through the rest of the solution. By carefully evaluating the exponential term, we ensure that the subsequent steps are based on a correct and simplified equation.
Step 3: Isolate the Variable Term
We want to get x² by itself on one side of the equation. To do this, we'll subtract 5 from both sides:
8 - 5 = x² + 5 - 5
3 = x²
Isolating the variable term is a fundamental algebraic technique used to solve equations. By subtracting 5 from both sides, we maintain the balance of the equation while moving closer to isolating x². This process simplifies the equation by reducing the number of terms and focusing on the variable we want to solve for. The goal is to get the variable term, in this case x², alone on one side so that we can apply further operations to find the value of x. This step demonstrates the importance of performing the same operation on both sides of an equation to preserve equality and progress towards the solution.
Step 4: Solve for x
To solve for x, we need to take the square root of both sides of the equation. Remember that when we take the square root, we need to consider both the positive and negative roots:
√(3) = √(x²)
So, x = ±√3
Solving for x by taking the square root introduces the concept of both positive and negative solutions, which is crucial for understanding quadratic equations. The square root of x² is not just x, but |x|, meaning x can be either √3 or -√3. This step highlights the importance of considering all possible solutions when dealing with square roots, as missing one solution would lead to an incomplete answer. The ± symbol is a concise way to represent both solutions, ensuring that we capture the full range of values that satisfy the equation. This step demonstrates a key principle in algebra: accounting for all possible outcomes when solving for a variable.
Step 5: Check the Solutions
It's always a good idea to check our solutions to make sure they work in the original equation. Let's plug x = √3 and x = -√3 back into log₂(x² + 5) = 3.
For x = √3:
log₂((√3)² + 5) = log₂(3 + 5) = log₂8 = 3 (This works!)
For x = -√3:
log₂((-√3)² + 5) = log₂(3 + 5) = log₂8 = 3 (This also works!)
Checking the solutions is a vital step in the problem-solving process, as it verifies the accuracy of our calculations and ensures that the solutions are valid within the original equation's context. By substituting both √3 and -√3 back into the original equation, we confirm that they both satisfy the logarithmic relationship. This step is especially important in logarithmic equations, where extraneous solutions can sometimes arise due to the domain restrictions of logarithmic functions. The verification process not only validates the answers but also reinforces understanding of the equation and the steps taken to solve it. This practice of checking solutions is a cornerstone of mathematical rigor and helps prevent errors.
The Solution Set
Both solutions check out, so our solution set is {√3, -√3}.
Key Takeaways
Solving logarithmic equations might seem daunting at first, but with a clear understanding of the relationship between logarithms and exponents, and by following a step-by-step approach, you can conquer any logarithmic challenge. Remember these key steps:
- Convert the logarithmic equation to exponential form.
- Simplify the exponential expression.
- Isolate the variable term.
- Solve for the variable, remembering to consider both positive and negative roots when taking square roots.
- Check your solutions in the original equation.
By mastering these steps, you'll be well-equipped to tackle a wide range of logarithmic equations. Keep practicing, and you'll become a logarithmic equation-solving superstar!
Practice Problems
Want to test your skills? Try solving these equations:
- log₃(2x + 1) = 2
- log₅(x² - 4) = 1
- log₄(3x - 2) = 2
Solving these practice problems will help solidify your understanding of the process and build your confidence in working with logarithmic equations. Each problem presents a slightly different scenario, allowing you to apply the steps we've discussed in various contexts. Remember to convert to exponential form, simplify, isolate the variable, solve, and check your solutions. The more you practice, the more natural these steps will become. Working through these problems will also help you identify any areas where you may need further review, ensuring a comprehensive understanding of how to solve logarithmic equations. So, grab a pencil and paper, and let's put your new skills to the test!
