Solving Equations Involving Logarithms: A Step-by-Step Guide

When faced with equations that involve logarithms, the key is to understand and apply the properties of logarithms properly. Let's explore a specific example to illustrate this process.

Understanding the Example: Solving for x

Given the equation: x -5.

Let's assume the equation is: 6x log 2 - 3x log 5 log 2 2 log 5. This simplifies to: 6x log 2 - 3x log 5 log 50.

Another formulation is: 3x log 4 - log 5 log 50. We can rewrite it as: x log (64/125) log 50.

Finally, we can solve for x as follows: x log 50 / log (64/125) -5..

Steps to Solve Logarithmic Equations

Step 1: Identify the Logarithmic Equation

Let's start with the equation: 6x log 2 - 3x log 5 log 50.

Step 2: Apply Logarithm Properties

First, combine the terms involving x on one side:

6x log 2 - 3x log 5 log 50

Factor out x:

6x log 2 - 3x log 5 log 50

Factor x out:

x (6 log 2 - 3 log 5) log 50

Step 3: Use Logarithm Properties to Simplify

Next, simplify the logarithmic terms:

6 log 2 log 2^6 log 64

3 log 5 log 5^3 log 125

Therefore, we have:

x (log 64 - log 125) log 50

Using the logarithm subtraction rule, log (64/125) log 50:

x log (64/125) log 50

Step 4: Solve for x

Now, divide both sides by log (64/125) to isolate x:

x log 50 / log (64/125)

Calculate the values:

x -5.

Understanding the Base of Logarithms

It's crucial to know the base of the logarithm in the equation. Themost common bases are:

Base 10 Logarithm (log)

Log without index is base 10 logarithm, which allows you to write any number as a power of 10. For example, log 1000 3 because 10^3 1000. For non-integral powers, like log 600 ≈ 2.778, this means 600 is 10 to the power of approximately 2.778.

Natural Logarithm (ln)

ln means loge, where e 2.718...

Base 2 Logarithm (log2)

Base 2 logarithm is important for powers of 2. For example, log2 512 9 because 2^9 512. This can also be found on the Windows calculator using the division rule: log2 512 ln 512 / ln 2 log 512 / log 2 9.

Conclusion

Solving logarithmic equations involves a clear understanding of logarithmic properties and the proper application of these rules. By breaking down the problem step-by-step, we can arrive at the solution efficiently.

Understanding the base of the logarithm is key, as different bases can be used in the same equation without changing the result. Whether you use base 10, natural logarithm, or any other base, the essential steps in solving the equation remain the same.