Solving Logarithmic Equations: A Step-by-Step Guide

Logarithmic equations can seem daunting at first, particularly when multiple logarithms appear in the same expression. However, by understanding and applying the properties of logarithms, solving such equations becomes much more manageable. This article will walk you through the process of solving equations like log_2 x1 - log_2 x log_2 4.

Understanding the Equation

The given equation is log_2 x1 - log_2 x log_2 4. This can be simplified using the property of logarithms that states log_a b - log_a c log_a (b/c).

Step 1: Applying the Logarithm Property

To simplify the left side of the equation, we apply this property:

log_2 x1 - log_2 x log_2 (x1/x)

This transforms the equation to:

log_2 (x1/x) log_2 4

Step 2: Equating the Arguments

Since both sides of the equation have the same logarithmic base (2), we can equate their arguments:

x1/x 4

Step 3: Solving for (x)

Next, we solve for x by cross-multiplying:

x1 4x

Rearranging the equation to isolate x:

1 4x - x1

1 3x

x 1/3

Step 4: Verifying the Solution

To confirm that x 1/3 is indeed a valid solution, we substitute it back into the original equation:

log_2 (1/3*1) - log_2 (1/3) log_2 4

Which simplifies to:

log_2 (4/3) - log_2 (1/3) log_2 4

Since 4/3 / (1/3) 4, the equation holds true:

log_2 4 log_2 4

This confirms that x 1/3 is the correct solution.

Key Takeaways

By understanding and applying the properties of logarithms, we can simplify complex equations and solve for unknown variables. The key properties used here are:

Remember to verify your solutions by substituting them back into the original equation to ensure they are valid.

Additional Examples

Here are a couple of additional examples to further reinforce the concept:

Example 1

Solve log_2 x1 - log_2 x log_2 4

1. Apply the logarithm property:

log_2 (x1/x) log_2 4

2. Equate the arguments:

x1/x 4

3. Solve for x:

1 3x

x 1/3

Example 2

Solve log_2 x1 - log_2 x log_2 4

1. Apply the logarithm property:

log_2 (x1/x) log_2 4

2. Equate the arguments:

x1/x 4

3. Solve for x:

1 3x

x 1/3

Conclusion

Mastering the solving of logarithmic equations is essential for any student of mathematics or anyone involved in fields that require quantitative analysis. By following the steps outlined in this article, you can confidently solve similar equations and apply this knowledge to more complex problems.