Solving Logarithmic Equations: Log10(2x 1) - Log10(3x-2) 1

In this article, we will explore the intricacies of solving a specific logarithmic equation, Log10(2x 1) - Log10(3x-2) 1. We will cover the key steps, properties used, and provide detailed explanations for each part of the solution. By the end, you will understand how to apply logarithmic properties to solve similar equations.

The Equation: Log10(2x 1) - Log10(3x-2) 1

Let's start by breaking down the given equation:

Log10(2x 1) - Log10(3x-2) 1

Our goal is to solve for x in the equation Log10(2x 1) - Log10(3x-2) 1.

Step-by-Step Solution

The first step is to simplify the left-hand side using the logarithmic property:

log_b(m) - log_b(n) log_b(frac{m}{n})

Applying this property, we get:

Log10(2x 1) - Log10(3x-2)  Log10(frac{2x 1}{3x-2})

Further simplifying, the equation becomes:

Log10(frac{2x 1}{3x-2})  1

Using the property that if log_b m n, then m b^n, we can rewrite the equation as:

frac{2x 1}{3x-2}  10

Further Simplification

Multiplying both sides of the equation by the denominator (3x - 2), we get:

2x   1  10(3x - 2)

Expanding the right-hand side, we have:

2x   1  3 - 20

Now, rearrange the equation to isolate x:

2x - 3  -20 - 1

-28x  -21

x  frac{21}{28}

x  frac{3}{4}

Thus, the solution to the equation is:

x  frac{3}{4}

Verification

To verify our solution, we substitute x 3/4 back into the original equation and check if it holds true:

Log10(2(frac{3}{4})   1) - Log10(3(frac{3}{4}) - 2)  1

Calculating each term:

Log10(frac{6}{4}   1) - Log10(frac{9}{4} - 2)  1

Log10(frac{10}{4}) - Log10(frac{1}{4})  1

Log10(2.5) - Log10(0.25)  1

We know that:

Log10(2.5)  Log10(2.5)

Log10(0.25)  Log10((frac{1}{4}))  Log10(frac{1}{10^2})  -2 * Log10(10)  -2

Therefore:

Log10(2.5) - (-2)  Log10(2.5)   2  1

Since the left and right sides are equal, our solution is verified.

Conclusion

In this article, we've discussed how to solve the logarithmic equation Log10(2x 1) - Log10(3x-2) 1 step-by-step. We used properties of logarithms such as the quotient rule and the definition of logarithmic equations to reach the solution. By understanding these properties and methods, you can solve similar logarithmic equations effectively.

Additional Reading:

Understanding Logarithms

Logarithm Properties Explained

Examples of Solving Logarithmic Equations

Stay tuned for more mathematical tutorials and insights!