Solving Logarithmic Equations: A Comprehensive Guide

This article provides a detailed step-by-step guide to solving complex logarithmic equations. By understanding the properties of logarithms, we can systematically solve equations involving logarithms. We'll focus on the specific problem of solving the equation 2 log10 3x-2 - log11 1 log10 2

Understanding Logarithms

Logarithms are a fundamental mathematical function that can be thought of as the inverse of exponential functions. The logarithm of a number to a specified base is the exponent to which the base must be raised to produce that number. For example, in the expression log10 1000, the base 10 must be raised to the power of 3 to produce 1000, thus log10 1000 3.

Basic Properties of Logarithms

There are several useful properties of logarithms that we need to remember when solving equations. These properties are crucial for simplifying and simplifying complex logarithmic expressions:

Product Rule: logb (m * n) logb m logb n Quotient Rule: logb (m / n) logb m - logb n Power Rule: logb (mn) n * logb m

These properties help us break down and simplify complex logarithmic expressions by using subtraction of logarithms, which is equivalent to division.

Solving a Specific Logarithmic Equation

The equation we will solve is 2 log10 3x-2 - log11 1 log10 2. Let's break this down step-by-step using the properties of logarithms:

Step 1: Simplify the Equation Using the Difference Property

According to the difference property of logarithms, logb m - logb n logb (m / n). Applying this to our equation:

2 log10 3x-2 - log11 1 log10 2

becomes

log10 (3x-2)2 - log11 1 log10 2

But since log11 1 0 (any log of 1 is 0), our equation simplifies to:

log10 (3x-2)2 log10 2

Step 2: Equate the Arguments

Since the bases are the same, the arguments must be equal:

(3x-2)2 2

Step 3: Solve for x

Now, we solve the quadratic equation for x:

(3x-2)2 2

Taking the square root of both sides gives:

3x-2 ±√2

Rearranging the terms gives:

3x 2 ± √2

Dividing by 3:

x (2 ± √2) / 3

Thus, the solutions are: x (2 √2) / 3 and x (2 - √2) / 3.

Cheking the Solution

It's important to check the solution in the original equation to ensure it is correct:

2 log10(3x-2) - log111 log102

Substitute x (2 √2) / 3 and x (2 - √2) / 3 to verify:

For x (2 √2) / 3, we calculate:

2 log10(3(2 √2)/3 - 2) - log111 2 log10(√2) - 0 log10(2)

And for x (2 - √2) / 3, we calculate:

2 log10(3(2 - √2)/3 - 2) - log111 2 log10(-√2) - 0 log10(2)

Note: The negative argument in the second case is not valid in the real number system since log10 of a negative number is not defined.