Solving Logarithmic Equations Ln(e^(ln(x))) + Ln(e^(ln(x^2))) = 2ln(8)

Jul 12, 2025 by ADMIN 71 views

Unraveling the Solution to a Logarithmic Equation: A Comprehensive Guide

Delving into the realm of logarithmic equations often presents a fascinating challenge, requiring a solid understanding of logarithmic properties and algebraic manipulation. In this comprehensive guide, we will embark on a step-by-step journey to unravel the solution to the equation ln(e^(ln(x))) + ln(e^(ln(x2))) = 2ln(8). Our exploration will not only provide the correct answer but also illuminate the underlying principles and techniques for tackling similar logarithmic problems. Let's embark on this mathematical adventure!

Understanding the Fundamentals of Logarithms

Before we dive into the intricacies of the equation, let's take a moment to refresh our understanding of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. The equation logb(a) = c signifies that b raised to the power of c equals a (b^c = a). Here, 'b' is the base of the logarithm, 'a' is the argument, and 'c' is the logarithm itself. Grasping this fundamental relationship is crucial for navigating the world of logarithmic equations.

Key Properties of Logarithms

To effectively manipulate and solve logarithmic equations, it is essential to be familiar with the key properties of logarithms. These properties serve as powerful tools in simplifying expressions and isolating the variable. Let's delve into some of these crucial properties:

Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as logb(mn) = logb(m) + logb(n).
Quotient Rule: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In mathematical notation, this is represented as 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. This can be written as logb(mp) = p * logb(m).
Change of Base Rule: This rule allows us to convert logarithms from one base to another. It states that logb(a) = logc(a) / logc(b), where 'c' is a new base.
Inverse Property: This property highlights the inverse relationship between logarithms and exponentiation. It states that blogb(x) = x and logb(bx) = x.

Solving Logarithmic Equations: A Step-by-Step Approach

Solving logarithmic equations involves a systematic approach that combines the properties of logarithms with algebraic manipulation. Here's a general strategy to guide you through the process:

Isolate the Logarithmic Term: Begin by isolating the logarithmic term on one side of the equation. This may involve algebraic operations such as addition, subtraction, multiplication, or division.
Convert to Exponential Form: Once the logarithmic term is isolated, convert the equation from logarithmic form to exponential form. This step utilizes the fundamental relationship between logarithms and exponentiation.
Solve for the Variable: After converting to exponential form, solve the resulting algebraic equation for the variable. This may involve techniques such as factoring, simplifying, or using the quadratic formula.
Check for Extraneous Solutions: It is crucial to check the solutions obtained to ensure they are valid. Logarithmic functions are only defined for positive arguments. Therefore, any solution that results in a negative or zero argument in the original equation must be discarded as an extraneous solution.

Unraveling the Equation: ln(e^(ln(x))) + ln(e^(ln(x2))) = 2ln(8)

Now that we have a solid foundation in logarithmic principles and problem-solving strategies, let's tackle the given equation: ln(e^(ln(x))) + ln(e^(ln(x2))) = 2ln(8). We will dissect this equation step-by-step, applying our knowledge to arrive at the true solution.

Step 1: Simplifying the Logarithmic Terms

The first step in solving this equation involves simplifying the logarithmic terms using the inverse property of logarithms. Recall that ln(e^u) = u. Applying this property to our equation, we get:

ln(x) + ln(x^2) = 2ln(8)

Step 2: Applying the Power Rule of Logarithms

Next, we can simplify the term ln(x^2) using the power rule of logarithms. The power rule states that ln(xp) = p * ln(x). Applying this rule, our equation becomes:

ln(x) + 2ln(x) = 2ln(8)

Step 3: Combining Like Terms

Now, we can combine the like terms on the left side of the equation:

3ln(x) = 2ln(8)

Step 4: Isolating the Logarithmic Term

To isolate the logarithmic term, we divide both sides of the equation by 3:

ln(x) = (2/3)ln(8)

Step 5: Applying the Power Rule in Reverse

We can rewrite the right side of the equation using the power rule in reverse. This gives us:

ln(x) = ln(8^(2/3))

Step 6: Simplifying the Exponent

Now, we simplify the exponent 8^(2/3). Recall that a fractional exponent represents a root. In this case, 8^(2/3) is the cube root of 8 squared, which is (8^(1/3))2 = 2^2 = 4. Therefore, our equation becomes:

ln(x) = ln(4)

Step 7: Solving for x

Since the natural logarithms on both sides of the equation are equal, we can conclude that the arguments must also be equal. Therefore:

x = 4

Step 8: Checking for Extraneous Solutions

Finally, we must check our solution to ensure it is not extraneous. Since ln(4) is defined, our solution x = 4 is valid.

The True Solution: x = 4

Therefore, the true solution to the equation ln(e^(ln(x))) + ln(e^(ln(x2))) = 2ln(8) is x = 4. This corresponds to option B in the given choices.

Conclusion: Mastering Logarithmic Equations

In this comprehensive guide, we have successfully unraveled the solution to the logarithmic equation ln(e^(ln(x))) + ln(e^(ln(x2))) = 2ln(8). By understanding the fundamental properties of logarithms, applying algebraic manipulation techniques, and carefully checking for extraneous solutions, we arrived at the correct answer, x = 4. Mastering the art of solving logarithmic equations is a valuable skill in mathematics, opening doors to a deeper understanding of various mathematical concepts and applications. As you continue your mathematical journey, remember that practice and a solid grasp of fundamental principles are the keys to success. Keep exploring, keep learning, and keep unraveling the mysteries of mathematics!