Problem 43 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x\) $$ \log _{2} 3+\log _{2} x=\log _{2} 5+\log _{2}(x-2) $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1 - Apply Logarithm Properties

Use the property of logarithms where \(\log_a m + \log_a n = \log_a (mn)\). This allows us to combine the terms on both sides of the equation: \(\log_{2}(3x) = \log_{2}(5(x-2))\).

Step 2 - Eliminate Logarithms by Equating Arguments

Since the base of the logarithms on both sides of the equation is the same, we can equate the arguments: \(3x = 5(x - 2)\).

Step 3 - Expand and Simplify

Distribute the \(5\) on the right side of the equation: \(3x = 5x - 10\).

Step 4 - Solve for \(x\)

Rearrange the equation to isolate \(x\): subtract \(5x\) from both sides to get \(-2x = -10\). Then, divide by \(-2\): \(x = 5\).

Step 5 - Verify the Solution

Check the solution by substituting \(x = 5\) back into the original equation: \(\log_{2} 3 + \log_{2} 5 = \log_{2} 5 + \log_{2} 3\), which simplifies to \(\log_{2} 15 = \log_{2} 15\), confirming \(x = 5\) is correct.

Key Concepts

Logarithm Properties

Logarithms come with several useful properties that help us simplify and solve equations. One of the fundamental properties is the product rule, which states that the sum of two logarithms with the same base is equal to the logarithm of their product:

• \( \log_a m + \log_a n = \log_a (mn) \)

In the context of this exercise, this property is used to combine the logarithmic terms on both sides of the equation.
When you have \( \log_2 3 + \log_2 x = \log_2 5 + \log_2 (x-2) \),
you can apply the product rule to rewrite it as \( \log_2 (3x) = \log_2 (5(x-2)) \).
This simplification makes it easier to solve the equation by focusing on the arguments within the logarithms.
Understanding these properties is crucial, as they offer powerful ways to break down complex expressions into simpler parts, making equations more approachable.

Solving Equations

Once the logarithmic properties have been applied to simplify the initial expression, solving the equation becomes straightforward. In this case, the equation \( \log_2 (3x) = \log_2 (5(x-2)) \) arises after using the properties. Since the logarithms on both sides have the same base,
we can eliminate the logarithms by equating their arguments directly:

• \( 3x = 5(x - 2) \)

This step significantly reduces the complexity of the problem, effectively transforming it from a logarithmic to a linear algebraic equation.
The key insight here is understanding that once you reach an equation \( \log_b A = \log_b B \), you can safely conclude \( A = B \).
This is because if logarithms with the same base are equal, their arguments must also be equal.

Algebraic Manipulation

With the logarithms eliminated, the task now involves solving a straightforward algebraic equation. Starting from \( 3x = 5(x-2) \),
the first step is to expand the right side by distributing the multiplication:

• \( 3x = 5x - 10 \)

Next, we rearrange the terms to isolate \(x\). Subtract \(5x\) from both sides, which results in:

• \( -2x = -10 \)

To solve for \(x\), divide each side by \(-2\), yielding:

• \( x = 5 \)

At this point, it's always wise to verify the solution by substituting back into the original equation. Plug \(x = 5\) into the original problem and confirm that both sides equal, ensuring no mistakes were made.
This verification process is crucial in developing robust problem-solving skills, as it helps confirm the solution is correct and reliable.