Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$ \log _{6}(2 x-3)=\log _{6} 12-\log _{6} 3 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{7}{2} \).

Step 1: Identify the Logarithmic Properties

The given equation is \( \log_{6}(2x-3) = \log_{6} 12 - \log_{6} 3 \). The right-hand side of the equation can be simplified using the logarithmic property: \( \log_{b} A - \log_{b} B = \log_{b}\left(\frac{A}{B}\right) \).

Step 2: Apply the Logarithmic Property

Apply the logarithmic property to the right-hand side: \( \log_{6}(2x-3) = \log_{6}\left(\frac{12}{3}\right) \). Simplify \( \frac{12}{3} \) to obtain \( 4 \), so the equation becomes \( \log_{6}(2x-3) = \log_{6}(4) \).

Step 3: Set the Arguments Equal

Since the logarithms have the same base, set the arguments equal: \( 2x - 3 = 4 \).

Step 4: Solve the Equation for x

Add 3 to both sides to get \( 2x = 7 \). Then, divide both sides by 2 to solve for \( x \): \( x = \frac{7}{2} \).

Step 5: Check the Solution

Substitute \( x = \frac{7}{2} \) back into the original equation to ensure it satisfies the equation. Calculate \( 2(\frac{7}{2}) - 3 \) which results in 4. Take \( \log_{6} 4 = \log_{6}(4) \), confirming that both sides agree.

Key Concepts

Logarithmic Properties

When solving logarithmic equations, understanding the properties of logarithms is crucial. These properties help in transforming and simplifying expressions, making it easier to solve the equations. One of the critical properties is the **difference of logarithms**:

• \( \log_{b} A - \log_{b} B = \log_{b}\left(\frac{A}{B}\right) \)

This property allows you to combine two logarithms into one, but only when they have the same base. Another essential property is the **product of logarithms**:

• \( \log_{b} (A \cdot B) = \log_{b} A + \log_{b} B \)

These properties simplify complex logarithmic equations and can transform a subtraction operation into a division inside a single logarithm. Utilization of these properties is often an initial step in solving logarithmic equations as we've shown in the provided exercise.

Base of Logarithm

The base of a logarithm is an essential concept that determines how the exponential function grows. The notation \( \log_{b} A \) represents the power to which the base \( b \) must be raised to yield \( A \). For instance, the base in our original problem is 6.
Understanding the base is important because the equation's properties work only when they have the same base. In our exercise, both sides had base 6, allowing us to set their arguments equal: this principle is due to the **logarithm equivalence rule**:

• If \( \log_{b}(M) = \log_{b}(N) \), then \( M = N \)

Logarithmic equations must be checked within their valid base range since they aren't defined for non-positive bases or for negative arguments. Consequently, ensuring consistent bases across terms simplifies solving processes and avoids calculation errors.

Solving Logarithmic Equations

Solving logarithmic equations involves several steps, starting with simplification using logarithmic properties and leading to isolation of the variable. Here’s the general approach taken in our exercise:

• **Simplify using logarithmic properties**: We used the subtraction property to combine \( \log_{6} 12 \) and \( \log_{6} 3 \) into \( \log_{6}(4) \).
• **Equate and solve**: With the equation \( \log_{6}(2x-3) = \log_{6}(4) \), we equate the arguments to get \( 2x-3 = 4 \).
• **Solve for the variable**: Solving the linear equation by adding 3 and dividing by 2 gives \( x = \frac{7}{2} \).

After solving, it’s essential to verify by substituting back into the original equation. This confirms the solution, ensuring it adheres to all conditions in the logarithmic domain. Completing these steps guarantees robustness and accuracy in solutions.