Solving Logarithmic Equations: A Detailed Guide

Logarithmic equations can often be challenging, but with the right approach and understanding of basic properties, they can be solved easily. In this article, we will go through the process of solving the equation log10 8x1 - log10 2x1 log12, step by step, to find the value of x.

Step 1: Apply Logarithm Properties

The first step is to use the logarithm property that states loga b - loga c loga (b/c). This simplifies our given equation:

log10 8x1 - log10 2x1 log12

Applying the property:

log10 [(8x1/2x1) log12]

Step 2: Simplify the Equation

Simplify the fraction inside the logarithm:

log10 (4) log12

Since the base of the logarithm on the left side simplifies to 4, we have:

4 x2

Solving this simple equation:

x2 4

x 1 and x -1

Step 3: Verify Solutions

Given the equation's nature and the properties used, we need to check if both solutions satisfy the original equation. However, the negative solution does not fit the context of the problem since logarithms of negative values are not defined in the real number system.

Therefore, the valid solution is:

x 1 and x 0.5

Conclusion

In summary, the equation log10 8x1 - log10 2x1 log12 has the solutions x 1 and x 0.5.

Further Exploration

Students and professionals in mathematics, science, and engineering often encounter logarithmic equations in various contexts. Understanding how to solve such equations is crucial for problem-solving and analytical thinking.

Keypoints to remember include:

The logarithm property loga b - loga c loga (b/c) How to manipulate and simplify expressions within logarithms The importance of checking solutions in the context of the problem