Solving the Equation logx1 logx-1 3 log 2 log 3 Using Logarithmic Properties

In this article, we will walk through a detailed solution to the logarithmic equation: logx1 logx-1 3 log 2 log 3. Understanding the key properties of logarithms will be essential in arriving at the final solution. We will apply these properties step-by-step to ensure clarity and a comprehensive understanding of the process.

Step-by-Step Solution

Step 1: Combine the left side of the equation using the property log a log b log ab.

logx1 logx-1 logx(1)(x - 1) logx(x - 1)

Step 2: Simplify the right side of the equation using the properties of logarithms.

3 log 2 log 3 log 23 log 3

log 8 log 3

log (8 log 3)

Therefore, the equation becomes:

logx(x - 1) log (8 log 3)

Final Step

Since the logarithms are equal, we can set the arguments equal to each other:

x - 1 24

x - 1 24

x2 - 1 24

x2 25

x plusmn;5

However, since we have a logarithmic term logx(x - 1), x - 1 must be positive. Therefore, eliminating x -5 because it would make the logarithm undefined:

x 5

Therefore, the only valid solution is x 5.

Conclusion

We have successfully solved the equation logx1 logx-1 3 log 2 log 3 using the properties of logarithms. By applying the rules log a log b log ab and log anb n log a log b, we arrived at the solution x 5. Understanding these fundamental logarithmic properties is crucial for solving complex equations like this one.

References

[1] Logarithm Rules

[2] Logarithms - Math Is Fun

[3] Logarithm Properties