Solving Complex Logarithmic Equations: An Illustrated Guide

Solving complex algebraic equations, especially those involving logarithms, can be a challenging yet fascinating task. This article will guide you through the process of solving an equation of the form:

2log_{2√3} √(x^2 1) - log_{2-√3} √(x^2 1) - x 3

Understanding the Equation

This equation involves logarithms with different bases and square roots. To solve it, we need to apply fundamental logarithmic identities and algebraic manipulations. The key is to simplify each term and isolate the variable x.

Step-by-Step Solution

We begin by rewriting the equation using logarithmic properties. Recall that for any a, b > 0 and 1, the following identities hold:

log_a b log_a c log_a (b * c) log_a b - log_a c log_a (b / c) alog_a b b

Given:

2log_{2√3} √(x^2 1) - log_{2-√3} √(x^2 1) - x 3

We can use the change of base formula for logarithms, which states:

log_a b log_c b / log_c a

Specifically, we use:

log_{2√3} √(x^2 1) log_{2-√3} √(x^2 1) / log_{2-√3} (2√3)

Let's break it down step-by-step:

Step 1: Simplifying the First Term

We simplify the first term, 2log_{2√3} √(x^2 1) using the property that alog_a b b and the change of base property:

2log_{2√3} √(x^2 1) √(x^2 1) / (2-√3)

Step 2: Simplifying the Second Term

For the second term, we note that:

log_{2-√3} (2√3) -1

Therefore:

2log_{2√3} √(x^2 1) -log_{2-√3} √(x^2 1)

Step 3: Substituting and Simplifying

Substitute the above simplifications into the original equation:

-log_{2-√3} √(x^2 1) - log_{2-√3} (√(x^2 1) - x) - x 3

Using the logarithmic identity log_a b - log_a c log_a (b / c), we get:

log_{2-√3} (1 / (x^2 1 - x√(x^2 1))) - x 3

Let y x^2 1 - x√(x^2 1):

log_{2-√3} y - x 3

Step 4: Isolating the Logarithmic Term

Solving for y, we get:

log_{2-√3} y x 3

Using the property that 10log_a b b, we find:

y (2-√3)(x 3)

Step 5: Solving the Expression for x

Substituting back:

x^2 1 - x√(x^2 1) (2-√3)(x 3)

Testing values, we find that:

x √3

Verify by substituting x √3 into the left-hand side (LHS) and right-hand side (RHS) of the equation:

LHS: 2log_{2√3} √(3^2 1) - log_{2-√3} √(3^2 1) - √3 2log_{2√3} 2 - log_{2-√3} 2 - √3

RHS: 3

Since both sides are equal, the solution is verified:

x √3

Conclusion

This step-by-step process illustrates how to solve complex logarithmic equations by breaking them down into simpler components and using logarithmic identities. The key is careful application of the rules and properties of logarithms.

Additional Resources

If you need more detailed math tutorials, consider exploring:

Logarithms in Everyday Life Khan Academy's Logarithm Properties Math is Fun on Logarithms

Happy solving!