Problem 51 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{6} \frac{x^{2}}{x+3} $$

Step-by-Step Solution

Answer

The expression is \( 2\log_6 x - \log_6 (x+3) \).

Step 1: Apply log division rule

The first step is to apply the logarithmic property for division, which states \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \). For the given expression, this becomes: \[ \log_6 \frac{x^2}{x+3} = \log_6 (x^2) - \log_6 (x+3) \] This property allows us to separate the division of terms inside the logarithm into a difference of two logarithms.

Step 2: Simplify the first logarithm using exponent rule

Now, apply the exponent rule for logarithms, which is \( \log_b (m^n) = n \log_b m \). For \( \log_6 (x^2) \), this rule becomes: \[ \log_6 (x^2) = 2 \log_6 x \] The exponent 2 is brought in front of the logarithm as a multiplier.

Step 3: Write the expression as a sum or difference of logarithms

Substitute the simplified form of the first term back into the expression from Step 1. The expression becomes: \[ 2 \log_6 x - \log_6 (x+3) \] This is the sum or difference representation of the original logarithmic expression.

Key Concepts

Logarithmic Properties

Logarithmic properties are fundamental tools that help simplify complex logarithmic expressions. Logarithms have rules similar to arithmetic operations, designed to make manipulation and simplification easier. There are several key properties of logarithms that are often used in algebra.

• **Product Rule:** This states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, it is expressed as: \( \log_b(m \cdot n) = \log_b m + \log_b n \).
• **Quotient Rule:** This is similar to the division rule and states that the logarithm of a quotient is the difference of the logarithms. Written as: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \).
• **Exponent Rule:** It says that the logarithm of a number raised to an exponent is simply the exponent times the logarithm of the base. Represented as: \( \log_b(m^n) = n \log_b m \).

These properties are powerful because they convert complicated multiplications, divisions, and exponentiations inside logarithms into simpler additions, subtractions, and multiplications outside the logarithm.

Division Rule for Logarithms

The division rule for logarithms, also known as the quotient rule, is a handy tool for breaking down complex expressions into simpler components. When you encounter a division inside a logarithm, the rule states you can write it as a subtraction instead.This division rule is expressed as:

• \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \)

This means wherever you have a fraction within a logarithm, it can be separated into the difference of two individual logarithms. This property simplifies calculations significantly, especially when dealing with variables and multi-step logarithmic expressions.In the original exercise, we used this rule on the expression \( \log_6 \frac{x^2}{x+3} \). By applying the division rule, this was rewritten as:

• \( \log_6 (x^2) - \log_6 (x+3) \)

This transformation helps in simplifying each part separately, making the overall problem more approachable.

Exponent Rule for Logarithms

The exponent rule for logarithms is a key principle used to simplify expressions where numbers inside a logarithm are raised to a power. This rule allows us to pull exponents out in front of a logarithm as a coefficient, making it easier to manage algebraic expressions.Mathematically, this rule is written as:

• \( \log_b(m^n) = n \log_b m \)

This shows that the exponent can come outside in front of the logarithm, turning what could be a complex nested power inside the function into a simple multiplication outside.In our exercise, we applied this rule to the term \( \log_6(x^2) \). By doing so, we expressed it as:

• \( 2 \log_6 x \)

This conversion is crucial for breaking down expressions and is particularly useful when expressions must be simplified to combine or manipulate them further. By mastering this rule, solving logarithmic problems becomes more straightforward and less daunting.