Problem 34 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{9}(4 x)-\log _{9}(x-3)+\log _{9}\left(x^{3}+1\right) $$

Step-by-Step Solution

Answer

\( \log_{9} \left( \frac{4x(x^{3}+1)}{x-3} \right) \)

Step 1: Apply the Subtraction Rule of Logarithms

The subtraction of two logs with the same base can be expressed as the division of their arguments. Thus, \( \log_{9}(4x) - \log_{9}(x-3) \) can be simplified using the rule: \( \log_{b} (a) - \log_{b} (c) = \log_{b} \left( \frac{a}{c} \right) \). Therefore, we write it as \( \log_{9} \left( \frac{4x}{x-3} \right) \).

Step 2: Apply the Addition Rule of Logarithms

The addition of two logs with the same base can be expressed as the multiplication of their arguments. Thus, \( \log_{9} \left( \frac{4x}{x-3} \right) + \log_{9}(x^{3}+1) \) can be simplified using the rule: \( \log_{b} (a) + \log_{b} (c) = \log_{b} (a \times c) \). Thus, we write it as \( \log_{9} \left( \frac{4x}{x-3} \times (x^{3}+1) \right) \).

Step 3: Simplify the Expression

Combine the expression obtained in Step 2 to form a single argument inside a single logarithm: \( \log_{9} \left( \frac{4x(x^{3}+1)}{x-3} \right) \).

Key Concepts

Subtraction Rule of Logarithms

The subtraction rule of logarithms is one of the fascinating aspects of log operations. It helps simplify expressions that involve the subtraction of two logarithms that share the same base. This rule states that when you subtract one log from another, with the same base, it can be rewritten as the log of a division. For instance, if you have

• \( \log_b(a) - \log_b(c) \)

It simplifies to:

• \( \log_b\left(\frac{a}{c}\right) \)

This means the operation of subtraction transforms into division, helping to streamline expressions with multiple logarithms.

Think of it as transforming the complexity of separate logarithms into the simpler perspective of one by dividing their content. This rule is particularly useful in handling lengthy mathematical expressions, simplifying what seems complicated at first glance.

Addition Rule of Logarithms

The addition rule of logarithms is another handy tool that mirrors the subtraction rule but aims at uniting logs through multiplication. When two logarithms with the same base are added together, this rule allows you to combine their arguments by multiplying them. The general formula is:

• \( \log_b(a) + \log_b(c) \)

This is equivalent to:

• \( \log_b(a \times c) \)

This rule is very straightforward and simplifies expressions by binding multiple logarithms into a singular expression through the act of multiplication.

In essence, the addition rule is about merging complexities into a single, cohesive component, simplifying equations involving logs into more manageable expressions. This is how, in our exercise, multiple logs combined into a singular expression using multiplication.

Logarithmic Simplification

Logarithmic simplification is the process of reducing complex logarithmic expressions into more straightforward single-log expressions. By doing this, the complexity of handling multiple logs is reduced. It often involves using the rules of subtraction and addition to achieve a more elegant and simple form.

In the given exercise, the expression started with multiple logs:

• \( \log_{9}(4x) - \log_{9}(x-3) + \log_{9}(x^3+1) \)

By applying both the subtraction and addition rules, the expression evolved, using division and multiplication, into:

• \( \log_{9}\left(\frac{4x(x^3+1)}{x-3}\right) \)

This simplification process involves transforming complex expressions into easier to evaluate forms, which are simpler to analyze or solve.

Simplification not only makes solving equations involving logs more efficient but also uncovers insights and relationships within mathematical functions more clearly.