Problem 26 - Algebra A Combined Function

Question

Solve each equation. $$ \log _{6}(x+2)-\log _{6} x=2 $$

Step-by-Step Solution

Answer

x = \(\frac{2}{35}\)

Step 1: Use the Logarithm Quotient Rule

The equation is \( \log_{6}(x + 2) - \log_{6} x = 2 \).Using the quotient rule of logarithms, which states that \( \log_{b}(a) - \log_{b}(c) = \log_{b}(\frac{a}{c}) \), we rewrite the equation as:\[\log_{6}\left( \frac{x + 2}{x} \right) = 2\]

Step 2: Convert the Logarithmic Equation to an Exponential Equation

To solve the equation \( \log_{6}\left( \frac{x + 2}{x} \right) = 2 \), convert it to its exponential form. The exponential form of a logarithmic equation \( \log_{b}(a) = c \) is \( a = b^{c} \).So, we have:\[\frac{x + 2}{x} = 6^{2}\]

Step 3: Simplify the Equation

Calculate the right-hand side of the equation:\[6^2 = 36\]Thus, the equation becomes:\[\frac{x + 2}{x} = 36\]

Step 4: Solve for x

Multiply both sides by \( x \) to eliminate the fraction:\[x + 2 = 36x\]Rearrange the equation to isolate \( x \):\[x - 36x = -2\]\[-35x = -2\]Solve for \( x \) by dividing both sides by \(-35\):\[x = \frac{2}{35}\]

Key Concepts

Understanding the Logarithm Quotient Rule

The logarithm quotient rule is a fundamental concept in mathematics that simplifies expressions involving differences of logarithms. It states that \( \log_{b}(a) - \log_{b}(c) = \log_{b}\left(\frac{a}{c}\right) \). This rule is incredibly valuable when simplifying or solving logarithmic equations.

In the given problem, we apply this rule to transform the equation \( \log_{6}(x + 2) - \log_{6} x = 2 \) into \( \log_{6}\left( \frac{x + 2}{x} \right) = 2 \). By converting the difference of logs into a single logarithm, we align the equation with a simpler form.

Applying the quotient rule helps in reducing the complexity of the expression, thus making it easier to solve step by step.

Solving with Exponential Equations

Once we simplify the logarithmic equation using the quotient rule, the next step is to convert it into an exponential form. The relationship between logarithms and exponents can be expressed as: if \( \log_{b}(a) = c \), then \( a = b^c \).

In this exercise, after applying the quotient rule, we have \( \log_{6}\left( \frac{x + 2}{x} \right) = 2 \). To solve for \( x \), convert this log equation to its exponential form: \( \frac{x + 2}{x} = 6^2 \).

This step is crucial because working with exponential equations can often be easier than working with logarithmic equations, especially when simplifying the problem further. Calculating \( 6^2 \) gives us 36, making the equation much simpler to handle: \( \frac{x + 2}{x} = 36 \).

Equation Simplification Process

After transforming the problem into a straightforward exponential equation, the simplification begins. Initially, the equation appears as \( \frac{x + 2}{x} = 36 \). Our goal now is to solve for \( x \).

First, eliminate the fraction by multiplying both sides by \( x \), yielding \( x + 2 = 36x \). This step is vital as it allows us to clear the fraction, simplifying the calculations.

Next, rearrange the equation to collect like terms: \( x - 36x = -2 \). This results in \( -35x = -2 \).

Finally, isolate \( x \) by dividing both sides by \(-35\), giving \( x = \frac{2}{35} \). What started as a complex logarithmic expression has been simplified through logical steps into a simple equation resulting in a precise value for \( x \). This process illustrates the power of algebraic manipulation in solving equations efficiently.