Solving Logarithmic Equations 2 Ln X - Ln (2x - 7) = Ln (9x) - Ln (x + 2)

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Logarithmic equations can seem daunting, but with a systematic approach, they can be solved effectively. In this guide, we'll tackle the equation 2 ln x - ln (2x - 7) = ln (9x) - ln (x + 2) step by step, providing a clear and detailed explanation of each step involved.

Understanding Logarithms

Before diving into the equation, it's crucial to grasp the fundamentals of logarithms. A logarithm is the inverse operation to exponentiation. In simpler terms, if we have an equation like b^y = x, then the logarithm of x to the base b is y, which can be written as log_b(x) = y. Natural logarithms, denoted as "ln", are logarithms to the base e, where e is an irrational number approximately equal to 2.71828.

Properties of Logarithms

Several key properties of logarithms are essential for solving logarithmic equations. These include:

  1. Product Rule: ln(ab) = ln(a) + ln(b)
  2. Quotient Rule: ln(a/b) = ln(a) - ln(b)
  3. Power Rule: ln(a^n) = n ln(a)

These properties allow us to manipulate logarithmic expressions and simplify equations. Understanding and applying these rules correctly is paramount to solving logarithmic equations. We will use these rules extensively in solving the given equation.

Preparing the Equation

The given equation is: 2 ln x - ln (2x - 7) = ln (9x) - ln (x + 2). Our first goal is to simplify the equation using the properties of logarithms. We'll start by applying the power rule to the first term on the left side of the equation. This means we can rewrite 2 ln x as ln(x^2). So the equation becomes:

ln(x^2) - ln(2x - 7) = ln(9x) - ln(x + 2)

Next, we'll use the quotient rule to combine the logarithms on each side of the equation. The quotient rule states that ln(a) - ln(b) = ln(a/b). Applying this rule to both sides, we get:

ln(x^2 / (2x - 7)) = ln(9x / (x + 2))

Now that we have a single logarithm on each side of the equation, we can proceed to the next step, which involves eliminating the logarithms.

Eliminating Logarithms

Since we have ln on both sides of the equation, we can eliminate the logarithms by setting the arguments (the expressions inside the logarithms) equal to each other. This is because if ln(a) = ln(b), then a = b. Therefore, we can write:

x^2 / (2x - 7) = 9x / (x + 2)

This step is crucial as it transforms the logarithmic equation into a simpler algebraic equation, which is easier to solve. Now, we have a rational equation that we can solve for x.

Solving the Algebraic Equation

To solve the equation x^2 / (2x - 7) = 9x / (x + 2), we first eliminate the fractions by cross-multiplying:

x^2(x + 2) = 9x(2x - 7)

Expanding both sides, we get:

x^3 + 2x^2 = 18x^2 - 63x

Now, we rearrange the equation to set it equal to zero. This involves moving all terms to one side of the equation:

x^3 + 2x^2 - 18x^2 + 63x = 0

Combining like terms, we simplify the equation to:

x^3 - 16x^2 + 63x = 0

Next, we factor out the common factor, which in this case is x:

x(x^2 - 16x + 63) = 0

Now, we have a quadratic equation inside the parentheses. We need to factor the quadratic equation x^2 - 16x + 63. We look for two numbers that multiply to 63 and add up to -16. These numbers are -7 and -9. So, we can factor the quadratic as:

x(x - 7)(x - 9) = 0

This gives us three possible solutions for x: x = 0, x = 7, and x = 9. However, we need to check these solutions in the original logarithmic equation to ensure they are valid.

Checking for Extraneous Solutions

When solving logarithmic equations, it's essential to check for extraneous solutions. These are solutions that satisfy the algebraic equation but do not satisfy the original logarithmic equation. Extraneous solutions can arise because the domain of a logarithmic function is restricted to positive real numbers. We need to make sure that the arguments of all logarithms in the original equation are positive for a solution to be valid.

The original equation is: 2 ln x - ln (2x - 7) = ln (9x) - ln (x + 2)

Let's check each potential solution:

  1. For x = 0: The terms ln x and ln(9x) are undefined because the logarithm of 0 is undefined. Therefore, x = 0 is an extraneous solution.
  2. For x = 7: We substitute x = 7 into the original equation:
    • 2 ln(7) - ln(2(7) - 7) = 2 ln(7) - ln(7)
    • ln(9(7)) - ln(7 + 2) = ln(63) - ln(9) Simplifying, we get:
    • ln(7^2) - ln(7) = ln(49) - ln(7) = ln(49/7) = ln(7)
    • ln(63) - ln(9) = ln(63/9) = ln(7) Since both sides are equal, x = 7 is a valid solution.
  3. For x = 9: We substitute x = 9 into the original equation:
    • 2 ln(9) - ln(2(9) - 7) = 2 ln(9) - ln(11)
    • ln(9(9)) - ln(9 + 2) = ln(81) - ln(11) Simplifying, we get:
    • ln(9^2) - ln(11) = ln(81) - ln(11) = ln(81/11)
    • ln(81) - ln(11) = ln(81/11) Since both sides are equal, x = 9 is also a valid solution.

Therefore, the valid solutions for the equation are x = 7 and x = 9.

Final Answer

In conclusion, by applying the properties of logarithms, simplifying the equation, and checking for extraneous solutions, we have found that the solutions to the logarithmic equation 2 ln x - ln (2x - 7) = ln (9x) - ln (x + 2) are x = 7 and x = 9.

Therefore, the correct choice is:

A. x = 7, 9