Problem 53 - Algebra and Trigonometry

Question

\(45-54\) . Use the Laws of Logarithms to combine the expression. $$ \frac{1}{3} \log (x+2)^{3}+\frac{1}{2}\left[\log x^{4}-\log \left(x^{2}-x-6\right)^{2}\right] $$

Step-by-Step Solution

Answer

\(\log \frac{(x+2)x^2}{x^2-x-6}\)

Step 1: Apply Power Rule to Logs

First, we apply the power rule of logarithms, which states that \( a \log b^c = c \log b^a \), to simplify each logarithmic term. In the expression \( \frac{1}{3} \log (x+2)^{3} \), we can simplify this to \( \log (x+2)^{3/3} = \log (x+2) \). For the term \( \frac{1}{2} [\log x^{4} - \log (x^{2}-x-6)^{2}] \), simplify \( \log x^{4} \) to \( 4 \log x \) and \( \log (x^{2}-x-6)^{2} \) to \( 2 \log (x^{2}-x-6) \).

Step 2: Simplify the Second Bracket

Now handle the expression inside the brackets: \( \log x^4 - \log (x^2 - x - 6)^2 \). Using the quotient rule for logarithms, which states \( \log a - \log b = \log \frac{a}{b} \), we write \( 4 \log x - 2 \log (x^2 - x - 6) \) as \( \log \frac{x^4}{(x^2 - x - 6)^2} \).

Step 3: Apply the Multiplication Factor

Multiply the simplified expression from Step 2 by \( \frac{1}{2} \): \( \frac{1}{2} \log \frac{x^4}{(x^2 - x - 6)^2} = \log \sqrt{\frac{x^4}{(x^2 - x - 6)^2}} = \log \frac{x^2}{x^2 - x - 6} \).

Step 4: Combine All Terms into a Single Logarithmic Expression

Combine the logs: \( \log (x+2) + \log \frac{x^2}{x^2 - x - 6} \). Using the product rule, \( \log a + \log b = \log (ab) \), this simplifies to \( \log \left[(x+2) \cdot \frac{x^2}{x^2 - x - 6}\right] \). Therefore, the combined form of the logarithmic expression is \( \log \frac{(x+2)x^2}{x^2-x-6} \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms simplifies expressions where exponents appear in a logarithm. This rule states that \( a \log b^c = c \log b^a \). In simple terms, you can "bring down" the exponent as a multiplier in front of the logarithm. For instance, in the case of the expression \( \frac{1}{3} \log (x+2)^3 \), you can move the 3 (which is the exponent of \((x+2)\)) to the front, turning it into \( 3 \cdot \frac{1}{3} \log (x+2) \). This simplifies further into \( \log (x+2) \) upon cancelling out the 3.
The power rule is particularly useful in breaking down complex expressions to make them easier to work with. Take for example \( \log x^4 \), through the power rule, it becomes \( 4\log x \). Similarly, \( \log (x^2-x-6)^2 \) becomes \( 2\log (x^2-x-6) \).
Understanding and applying the power rule is the first step in managing and simplifying logarithmic expressions efficiently.

Quotient Rule of Logarithms

The quotient rule of logarithms helps simplify scenarios in which two logs are subtracted from each other. This rule states: \( \log a - \log b = \log \frac{a}{b} \). Fundamentally, it transforms a subtraction of two logarithms into a single logarithm by dividing the inside parts.
Consider the expression \( \log x^4 - \log (x^2-x-6)^2 \). By applying the quotient rule here, we change this to a single logarithm: \( \log \frac{x^4}{(x^2-x-6)^2} \).
The beauty of this rule lies in its ability to simplify multi-logarithmic expressions into a form that is easier to handle. Just remember that subtraction between two logarithms translates into a division within a single logarithm.

Product Rule of Logarithms

The product rule of logarithms goes hand in hand when adding two logarithms in an expression. It proposes that \( \log a + \log b = \log (ab) \), meaning an addition of two logs becomes a multiplication within a single log.
Let's observe the computation: combining \( \log (x+2) \) and \( \log \frac{x^2}{x^2-x-6} \) results in \( \log \, ((x+2) \cdot \frac{x^2}{x^2-x-6}) \).
The product rule plays a significant role in merging multiple logarithms into a singular expression. This final combination offers a more tidy and comprehensive expression, benefiting calculations and deeper understanding of the relationships within the mathematical problem.