The solution to 4log4(x+8)=4^2 is x = 4. When solving logarithmic equations, it’s crucial to remember the properties of logarithms. Begin by rewriting the equation in exponential form to simplify the calculation process. By isolating the variable x step by step, you can arrive at the final answer efficiently. Let’s delve deeper into unraveling the mystery behind logarithmic equations like 4log4(x+8)=4^2 to equip ourselves with the necessary problem-solving skills. Understanding the fundamental principles will undoubtedly lead you to successful solutions.
Exploring the Solution to 4log4(x+8) = 42
Welcome, young mathematicians! Today, we are going to embark on an exciting journey to unravel the mystery behind the equation 4log4(x+8) = 42. Don’t worry if those numbers and symbols seem intimidating at first glance. By the end of this adventure, you’ll have a solid understanding of how to solve this equation like a math wizard!
Understanding the Basics
Let’s break down the equation into smaller pieces to make it easier to digest. The term log4(x+8) might look complex, but it simply represents a logarithmic function where the base is 4. The number 42, on the other hand, is equal to 16.
So, in simpler terms, our equation can be rewritten as 4log4(x+8) = 16. Now that we’ve demystified the equation a bit, let’s move on to the next step of our journey – solving it!
Solving the Equation Step by Step
To find the solution to our equation, we need to isolate the variable x. We’ll do this by following a series of steps:
Step 1: Rewrite the Equation
Our equation is 4log4(x+8) = 16. We can rewrite this equation in exponential form to make it easier to work with. Remember, in logarithmic form, logb(x) = y can be expressed as by = x.
Applying this concept to our equation, we get 416 = x + 8.
Step 2: Simplify the Exponential Expression
Now, let’s simplify the exponential expression 416. Since 4 = 22, we can rewrite 416 as (22)16 = 232.
Therefore, our equation becomes 232 = x + 8.
Step 3: Solve for x
To solve for x, we need to isolate it on one side of the equation. Subtract 8 from both sides to get x = 232 – 8.
By simplifying the expression, we find that the solution to 4log4(x+8) = 42 is x = 4294967292.
Visualizing the Solution
It’s always helpful to visualize the solution to better understand the outcome of our mathematical journey. Imagine you have a magical box that takes any number you put into it, multiplies that number by 4, and then adds 8 to the result. If you put the number 4294967292 into this magical box, what would come out?
Let’s calculate: 4 * 4294967292 + 8 = 17179869176 + 8 = 17179869184.
So, the number that comes out of our magical box when we input x = 4294967292 is 17179869184. Isn’t it fascinating how math helps us solve puzzles and unlock hidden treasures?
Congratulations, young mathematicians! You’ve successfully navigated through the twists and turns of the equation 4log4(x+8) = 42 and emerged victorious with the solution x = 4294967292. Remember, math may sometimes seem like a daunting adventure, but with the right tools and a curious mind, you can conquer any mathematical challenge that comes your way. Keep exploring, keep learning, and let the magic of math guide you on your journey towards becoming a math wizard!
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Frequently Asked Questions
How can I solve the equation 4log₄(x+8) = 4²?
To solve the equation 4log₄(x+8) = 4², you first need to isolate the logarithmic term. Rewrite 4² as 16, then convert the equation to its exponential form so that 4⁴ = x+8. Simplify to get x = 240.
What steps should I follow to find the solution for 4log₄(x+8) = 4²?
To find the solution to 4log₄(x+8) = 4², isolate the logarithmic term first by converting 4² to 16. Rewrite the equation in exponential form, solve for x, and double-check your calculation to ensure accuracy.
Can you explain the process of solving 4log₄(x+8) = 4² in a step-by-step manner?
To solve 4log₄(x+8) = 4², begin by rewriting 4² as 16. Then, convert the logarithmic equation to its exponential form, which gives 4⁴ = x+8. Simplify this equation to find x = 240 as the final solution.
Final Thoughts
In conclusion, solving the equation 4log4(x+8) = 4^2 involves applying the properties of logarithms to simplify the equation. By converting log4 to its exponential form, the solution can be found. Taking the necessary steps to isolate x will lead to the final answer. Remember to check your solution to ensure it is valid for the given equation.
