Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$ \log (x+2)-\log x=2 \log 4 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{2}{15}\).

Step 1: Use Logarithm Properties

First, we apply the property of logarithms which states that \(\log a - \log b = \log\left(\frac{a}{b}\right)\). Applying this to \(\log (x+2) - \log x\) yields \(\log\left(\frac{x+2}{x}\right)\). So the equation becomes \(\log\left(\frac{x+2}{x}\right) = 2\log 4\).

Step 2: Simplify the Right Side

We use another property of logarithms which is \(n \log a = \log a^n\). Thus, \(2\log 4\) becomes \(\log 4^2 = \log 16\). So, the equation simplifies to \(\log\left(\frac{x+2}{x}\right) = \log 16\).

Step 3: Equate the Arguments

Since the two expressions are equal and the logarithms are to the base 10, we equate the arguments: \(\frac{x+2}{x} = 16\).

Step 4: Solve the Rational Equation

Multiply both sides by \(x\) to eliminate the fraction: \(x+2 = 16x\). Simplifying this, we get: \(2 = 16x - x\), leading to \(2 = 15x\).

Step 5: Solve for \(x\)

Divide both sides by 15 to isolate \(x\): \(x = \frac{2}{15}\).

Key Concepts

Logarithm Properties

Understanding logarithm properties is crucial when solving equations that involve logarithms. These properties help simplify expressions and make the equations easier to solve. One important property of logarithms is the subtraction rule, which states:

• \( \log a - \log b = \log\left(\frac{a}{b}\right) \)

This rule allows us to combine two logarithm expressions into a single logarithm that involves division. In the given problem, we used this property to rewrite \( \log (x+2) - \log x \) as \( \log\left(\frac{x+2}{x}\right) \).
Another vital property is the power rule of logarithms. It states:

• \( n \log a = \log a^n \)

This property enables us to move a coefficient in front of the logarithm to a power inside the logarithm. For instance, \( 2\log 4 \) was transformed into \( \log 4^2 = \log 16 \). By using these properties effectively, equations that initially seem complex can become manageable.

Solving Equations

Once logarithmic expressions are simplified using logarithm properties, the next step is to solve the resulting equation. In the exercise, we arrived at the equation \( \log\left(\frac{x+2}{x}\right) = \log 16 \). When both sides of an equation have logarithms with the same base, we can equate the arguments directly:

• \( \frac{x+2}{x} = 16 \)

This elimination of the logarithm functions significantly simplifies the equation.
Next, it is often helpful to clear any fractions by multiplying all terms by the denominator. Here, multiplying both sides by \( x \) removed the fraction, leading to: \( x+2 = 16x \). The goal is to isolate \( x \), which involves algebraic steps further described in the following section.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to find the value of the unknown variable. In our example, once we reached \( x + 2 = 16x \), the key was to group similar terms together. Subtract \( x \) from both sides to keep all \( x \) terms on one side, resulting in:

• \( 2 = 16x - x \)
• which simplifies to \( 2 = 15x \)

To solve for \( x \), we divide both sides by the coefficient of \( x \), which is 15:

• \( x = \frac{2}{15} \)

This simple division isolates the variable \( x \), providing the solution. Understanding how to manipulate equations with various mathematical operations, such as addition, subtraction, multiplication, and division, is essential. This knowledge allows one to solve more complex algebraic and logarithmic equations effectively.