Solving Complex Logarithmic Equations: A Comprehensive Guide

Logarithmic equations can be challenging to solve, but understanding the rules of logarithms makes the process much easier. In this article, we will explore how to solve a specific equation and provide a step-by-step guide that even a Google SEOer can appreciate. We will focus on both natural and common logarithms and highlight the key rules and steps involved.

Understanding the Problem:

The given equation is:

log x2 log x-2 log x23 - 2

Throughout this problem, we use the natural logarithm (loge), which is the base e logarithm.

Solving the Equation

First, we use the property that log a log b log (ab):

log (x2 * x-2) log (x23 - 2)

Since x2 * x-2 1, the equation simplifies to:

log 1 log (x23 - 2)

Because log 1 0, we have:

0 log (x23 - 2)

Converting from logarithmic form to exponential form (where log b a implies b 10a):

x23 - 2 1

Adding 2 to both sides:

x23 3

Now, we solve for x:

x 3(1/23)

Replacing with Common Logarithms

Let's solve the same problem using common logarithms (log10), where the base is 10:

log (x2) log (x-2) log (x23) - 2

Using the property log a log b log (ab):

log (x2 * x-2) log (x23 - 2)

Since x2 * x-2 1, the equation simplifies to:

log 1 log (x23 - 2)

Because log 1 0, we have:

0 log (x23 - 2)

Converting from logarithmic form to exponential form:

x23 - 2 100

Since 100 1, we have:

x23 - 2 1

Adding 2 to both sides:

x23 3

Now, we solve for x:

x 3(1/23)

Conclusion

In this article, we solved a complex logarithmic equation using both natural and common logarithms. The key steps included combining logarithmic terms, simplifying the equation, and solving for the variable. By understanding the properties of logarithms and converting between logarithmic and exponential forms, we were able to find the solution.

Keywords:

logarithmic equations natural logarithms exponential form