Simplify/Condense 4 log base 3 of x-1/3 log base 3 of x+6

Solution:

Step 1: Break down each term

Step 1.1

Apply the power rule of logarithms to $4 \log_{3}(x)$ to bring the coefficient inside the logarithm as an exponent: $\log_{3}(x^4) - \frac{1}{3} \cdot \log_{3}(x + 6)$

Step 1.2

Address the term $- \frac{1}{3} \log_{3}(x + 6)$.

Step 1.2.1

Rearrange the terms to maintain the order: $\log_{3}(x^4) - \frac{1}{3} \log_{3}(x + 6)$

Step 1.2.2

Apply the power rule of logarithms to $\frac{1}{3} \log_{3}(x + 6)$ to bring the coefficient inside the logarithm as an exponent: $\log_{3}(x^4) - \log_{3}((x + 6)^{\frac{1}{3}})$

Step 2: Combine using logarithm properties

Use the quotient rule for logarithms, which states that $\log_{b}(x) - \log_{b}(y) = \log_{b}\left(\frac{x}{y}\right)$, to combine the two terms: $\log_{3}\left(\frac{x^4}{(x + 6)^{\frac{1}{3}}}\right)$

Knowledge Notes:

The problem involves simplifying a logarithmic expression using the properties of logarithms. The key properties used in the solution are:

1. Power Rule: For any real number $a$, base $b$ (where $b > 0$ and $b \neq 1$), and positive number $x$, the power rule of logarithms states that $a \cdot \log_{b}(x) = \log_{b}(x^a)$.

2. Quotient Rule: For any positive numbers $x$ and $y$, and base $b$ (where $b > 0$ and $b \neq 1$), the quotient rule of logarithms states that $\log_{b}(x) - \log_{b}(y) = \log_{b}\left(\frac{x}{y}\right)$.

In the given problem, these properties are used to combine terms and simplify the expression. The power rule is applied first to move the coefficients inside the logarithms as exponents. Then, the quotient rule is used to combine the logarithmic terms into a single term by expressing them as a logarithm of a quotient. The final expression represents the original problem in its simplest form.