Solving Systems of Logarithmic Equations: A Comprehensive Guide

Logarithmic equations can be quite tricky, but with a systematic approach, they become manageable. In this article, we will walk through the steps to solve the following system of equations:

1. Simplification and Basic Logarithmic Properties

To tackle the system of equations, we first recognize and utilize some fundamental properties of logarithms. The given system is:

2 cdot log3y2 cdot log3x log2y-1log2x

Let's break these down step by step.

2. Solving the First Equation

We start with the first equation:

2 cdot log3y2 cdot log3x

By dividing both sides by 2, we get:

log3ylog332 cdot log3x

Using the property logabmm cdot logab, we can rewrite this as:

log3ylog39 cdotlog3x

This implies:

y3 cdotsqrt{x}

3. Solving the Second Equation

Now, we move to the second equation:

log2y-11 cdotlog2x

By recognizing that log221, we can rewrite the equation as:

log2y-log22log2x

Combining the logarithms using logab-logaclogabc, we get:

log2y2log2x

This implies:

y-12x

Thus, we have:

y2x 1

4. Substitution and Solving the Quadratic Equation

Now, we have two expressions for y:

y 3 · √{x}

and

y 2x 1

By setting these equal to each other, we get:

y 3 · √{x} similarly y 2x 1

Setting these equal:

3 cdotsqrt{x}2x 1

To eliminate the square root, we square both sides:

xserx 4x^2 - 5x 1 0

5. Applying the Quadratic Formula

Using the quadratic formula, we can solve the quadratic equation 4x^2 - 5x 1 0:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

With a 4, b -5, and c 1, we find:

x frac{5 pm sqrt{25 - 16}}{8} frac{5 pm sqrt{9}}{8} frac{5 pm 3}{8}

This gives us two possible values for x: x 1 and x 0.25

6. Finding Corresponding y Values

Next, we find the corresponding y values for these x values:

For x 1: y 3 · sqrt{1} 3 For x 0.25: y 3 · sqrt{0.25} 3 cdot frac{1}{2} frac{3}{2}

Thus, the solutions to the system of equations are:

x 1, y 3 x 0.25, y 1.5

Conclusion: Through the systematic application of logarithmic properties and solving quadratic equations, we have successfully solved the given system of equations. Understanding these steps can significantly enhance your problem-solving skills in advanced mathematics.

References

Lial, M. L., Miller, C. D. (2017). College Algebra and Trigonometry. Pearson. Stewart, J., Redlin, L., Watson, S. (2015). College Algebra. Cengage Learning.