Problem 52 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{3} \frac{(x+5)^{2}}{x} $$

Step-by-Step Solution

Answer

The expression is \(2 \cdot \log_3 (x+5) - \log_3 x\).

Step 1: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b \frac{M}{N} = \log_b M - \log_b N \). Use this rule to separate the fraction:\[ \log_3 \frac{(x+5)^2}{x} = \log_3 (x+5)^2 - \log_3 x \]

Step 2: Apply the Power Rule for Logarithms

The power rule for logarithms states that \( \log_b (M^n) = n \cdot \log_b M \). Use this rule on the first term obtained in Step 1:\[ \log_3 (x+5)^2 = 2 \cdot \log_3 (x+5) \]

Step 3: Combine Results from Steps 1 and 2

Substitute the result from Step 2 back into the expression obtained in Step 1. Now you have the expression written as a sum or difference of logarithms:\[ \log_3 \frac{(x+5)^2}{x} = 2 \cdot \log_3 (x+5) - \log_3 x \]

Key Concepts

Quotient Rule for Logarithms

Logarithms offer several rules to simplify expressions. One such useful rule is the quotient rule. This rule allows us to handle logarithmic expressions that involve divisions. It states that when you take the logarithm of a division, it can be expressed as the difference of two logarithms. Mathematically, this is written as:

• \( \log_b \frac{M}{N} = \log_b M - \log_b N\)

Here, \(M\) and \(N\) are positive numbers.
Let's apply this to the exercise. The original expression \(\log_3 \frac{(x+5)^2}{x}\) can be rewritten using the quotient rule. It becomes:

• \( \log_3 (x+5)^2 - \log_3 x\)

Effectively, this separates the division into two separate logarithmic terms. Understanding this rule is crucial for simplifying complex logarithmic expressions in a step-by-step manner.

Power Rule for Logarithms

Another powerful tool in manipulating logarithmic expressions is the power rule. This rule helps manage exponents inside logarithms. The power rule states that multiplying the logarithm of a number raised to a power is equal to that power times the logarithm of the base number:

• \( \log_b (M^n) = n \cdot \log_b M\)

Where \(n\) is any real number, and \(M\) is positive.
In our exercise, we encounter \((x+5)^2\). Using the power rule, we rewrite this expression as:

• \( 2 \cdot \log_3 (x+5)\)

This makes it easier to work with and understand the underlying logarithmic relationships. Recognizing when and how to use the power rule is essential to simplifying expressions deeply and correctly.

Properties of Logarithms

Logarithms are governed by several fundamental properties that enable us to transform and interpret expressions efficiently. Besides the quotient and power rules, there are other essential properties:

• Product Rule: \(\log_b (M \cdot N) = \log_b M + \log_b N\)
• Change of Base Formula: \(\log_b M = \frac{\log_k M}{\log_k b}\)

Each property serves a unique purpose and becomes a valuable tool in solving logarithmic problems.
By using these rules together, like in our exercise, we can simplify complex logarithmic expressions step-by-step. This makes solving them more manageable. The mastery of these properties makes it easier for students to tackle logarithms in their mathematical journey.