Solving the Logarithmic Equation log?(x2 - 1) 0: A Comprehensive Guide

When faced with a logarithmic equation like log2(x2 - 1) 0, it is essential to break down the problem step by step. This article will guide you through the process of solving such equations, providing a clear and structured approach to ensure you understand each intermediate step.

Understanding the Logarithmic Equation

The fundamental property of logarithms is that logb(y) 0 implies y b0. Applying this to our equation, we can rewrite the given equation as:

log2(x2 - 1) 0

Using the property mentioned, we get:

x2 - 1 20

This simplifies to:

x2-11

Adding 1 to both sides gives us:

x22

Solving for x2

Next, we need to solve for x2. This can be done by adding 1 to both sides. After doing so, we obtain:

x^22

Taking the Square Root

To find x, we take the square root of both sides. The square root of 2 is either the positive or negative root:

x ±√2

Checking for Validity

It is crucial to check the validity of our solutions. For a logarithm to be defined, its argument must be positive. In this case, we need to ensure that:

x2-10

Substituting x ±√2 back into the equation, we get:

2-12-11

Since 1 is greater than 0, both x √2 and x -√2 are valid solutions.

Conclusion

Thus, the solutions to the equation log2(x2 - 1) 0 are:

x √2 and x -√2

Both solutions are valid because they ensure that the argument of the logarithm, x2 - 1, is positive.

Further Reading

If you are interested in additional practice or deeper understanding of logarithmic equations, consider exploring related topics such as:

Algebraic Methods for Solving Logarithmic Equations Properties of Logarithms Logarithmic and Exponential Functions

By mastering the steps and techniques presented here, you will be well-equipped to tackle similar problems involving logarithms and algebraic expressions.