Solving The Logarithmic Equation: Find The Value Of X

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Let's dive into solving the logarithmic equation 3extln2+extln8=2extln(4x)3 ext{ln} 2 + ext{ln} 8 = 2 ext{ln} (4x). This problem combines several logarithmic properties that we'll need to unpack step by step. Understanding these properties is crucial for success in mathematics, especially when dealing with more complex equations. The journey to the solution involves simplifying logarithmic expressions, applying power rules, product rules, and finally, solving for the unknown variable. Let's embark on this mathematical adventure together and discover the true value of x.

Understanding the Basics of Logarithms

Before we jump into the solution, it’s essential to understand what logarithms are and their fundamental properties. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, if we have an equation like by=xb^y = x, we can express this in logarithmic form as $ ext{log}_b x = y$. Here, bb is the base of the logarithm, xx is the argument, and yy is the exponent. When we talk about the natural logarithm, denoted as $ ext{ln}$, we are referring to a logarithm with base ee, where ee is an irrational number approximately equal to 2.71828. The natural logarithm is widely used in calculus and various scientific applications, making it a cornerstone of advanced mathematical studies. To effectively solve logarithmic equations, we need to be familiar with several key properties. These properties act as tools in our mathematical toolkit, allowing us to manipulate and simplify expressions to reach a solution. Let’s explore these properties, which include the power rule, product rule, quotient rule, and the change of base formula.

Key Logarithmic Properties

  1. Power Rule: The power rule states that $ ext{log}_b(x^p) = p ext{log}_b(x)$. This rule allows us to move exponents from inside the logarithm to outside as a coefficient, which is incredibly useful in simplifying equations.
  2. Product Rule: The product rule tells us that $ ext{log}_b(xy) = ext{log}_b(x) + ext{log}_b(y)$. This rule allows us to combine the logarithms of products into the sum of individual logarithms, which is particularly useful when we need to expand logarithmic expressions.
  3. Quotient Rule: The quotient rule states that $ ext{log}_b( rac{x}{y}) = ext{log}_b(x) - ext{log}_b(y)$. This rule is similar to the product rule but applies to division. It’s used to separate the logarithm of a quotient into the difference of logarithms.
  4. Change of Base Formula: Although not directly used in this problem, it’s worth mentioning the change of base formula, which states that $ ext{log}_b(a) = rac{ ext{log}_c(a)}{ ext{log}_c(b)}$. This formula allows us to convert logarithms from one base to another, which is essential when using calculators that may not support a specific base.

Understanding these properties is the first step in tackling logarithmic equations. With these rules in mind, let's break down our original equation and apply these properties to simplify and solve it.

Step-by-Step Solution

Now, let's get our hands dirty with the equation 3extln2+extln8=2extln(4x)3 ext{ln} 2 + ext{ln} 8 = 2 ext{ln} (4x). Our goal here is to isolate x and find its true value. We’ll do this by strategically applying the logarithmic properties we discussed earlier. The process might seem intricate at first, but breaking it down into manageable steps makes it quite straightforward.

1. Simplify Using the Power Rule

Our first step is to simplify the equation using the power rule. Recall that the power rule states $ ext{log}_b(x^p) = p ext{log}_b(x)$. Applying this rule in reverse, we can rewrite the terms on both sides of the equation. On the left side, we have 3extln23 ext{ln} 2, which can be rewritten as $ ext{ln}(2^3)$. Similarly, on the right side, we have 2extln(4x)2 ext{ln} (4x), which can be rewritten as $ ext{ln}((4x)^2)$. Our equation now looks like this:

extln(23)+extln8=extln((4x)2) ext{ln}(2^3) + ext{ln} 8 = ext{ln}((4x)^2)

2. Further Simplification

Next, we simplify 232^3 to 8. So, the left side becomes $ ext{ln} 8 + ext{ln} 8$. On the right side, we simplify (4x)2(4x)^2 to 16x216x^2. The equation now is:

extln8+extln8=extln(16x2) ext{ln} 8 + ext{ln} 8 = ext{ln}(16x^2)

3. Apply the Product Rule

Now we use the product rule, which states that $ extlog}_b(xy) = ext{log}_b(x) + ext{log}_b(y)$. We can combine the two logarithmic terms on the left side $ ext{ln 8 + ext{ln} 8$ becomes $ ext{ln}(8 imes 8)$, which simplifies to $ ext{ln} 64$. Our equation now looks like this:

extln64=extln(16x2) ext{ln} 64 = ext{ln}(16x^2)

4. Eliminate the Logarithms

Since we have a natural logarithm on both sides of the equation, we can eliminate them by recognizing that if $ ext{ln} a = ext{ln} b$, then a=ba = b. Therefore, we can equate the arguments of the logarithms:

64=16x264 = 16x^2

5. Solve for x

Now, it’s a straightforward algebraic step to solve for x. Divide both sides by 16:

rac{64}{16} = x^2

4=x24 = x^2

Taking the square root of both sides gives us two possible solutions: x=2x = 2 and x=−2x = -2. However, we need to be cautious and check these solutions in the original equation because logarithms are not defined for negative arguments.

6. Check the Solutions

Let’s plug x=2x = 2 back into the original equation:

3extln2+extln8=2extln(4imes2)3 ext{ln} 2 + ext{ln} 8 = 2 ext{ln} (4 imes 2)

3extln2+extln8=2extln83 ext{ln} 2 + ext{ln} 8 = 2 ext{ln} 8

extln(23)+extln8=extln(82) ext{ln} (2^3) + ext{ln} 8 = ext{ln} (8^2)

extln8+extln8=extln64 ext{ln} 8 + ext{ln} 8 = ext{ln} 64

extln(8imes8)=extln64 ext{ln} (8 imes 8) = ext{ln} 64

extln64=extln64 ext{ln} 64 = ext{ln} 64

This solution is valid.

Now, let’s check x=−2x = -2. Plugging it into the original equation, we get:

3extln2+extln8=2extln(4imes−2)3 ext{ln} 2 + ext{ln} 8 = 2 ext{ln} (4 imes -2)

Since we have $ ext{ln} (-8)$, and the natural logarithm is not defined for negative numbers, x=−2x = -2 is not a valid solution. Therefore, the only valid solution is x=2x = 2.

Conclusion

After carefully applying logarithmic properties and checking our solutions, we have found that the true solution to the equation 3extln2+extln8=2extln(4x)3 ext{ln} 2 + ext{ln} 8 = 2 ext{ln} (4x) is x=2x = 2. This journey through the problem has highlighted the importance of understanding logarithmic properties and the necessity of checking solutions in the original equation to avoid extraneous roots. Logarithmic equations may seem daunting at first, but with a clear understanding of the properties and a systematic approach, they can be solved effectively. Practice is key, and each solved problem enhances our mathematical toolkit, preparing us for more complex challenges in the future. Remember, mathematics is not just about finding the right answer; it’s about the process of logical thinking and problem-solving. Keep exploring, keep practicing, and you’ll continue to enhance your mathematical prowess. For further learning and resources on logarithmic equations and properties, you can visit Khan Academy's Logarithm Section. This trusted website provides a wealth of information and practice exercises to deepen your understanding of this essential mathematical topic.