Alright! Let's carefully expand the logarithmic expressions using the logarithm rules you listed. I'll demonstrate with an example and explain each step in detail.
Let's expand the following logarithmic expression:
[tex]$ \log_a \left(\frac{x^3 y}{\sqrt{x + y}}\right) $[/tex]
1. Break down the logarithm of a quotient:
[tex]\[
\log_a \left(\frac{x^3 y}{\sqrt{x + y}}\right) = \log_a (x^3 y) - \log_a (\sqrt{x + y})
\][/tex]
This comes from the rule:
[tex]\[
\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y
\][/tex]
2. Break down the logarithm of the product:
[tex]\[
\log_a (x^3 y) = \log_a x^3 + \log_a y
\][/tex]
This uses the rule:
[tex]\[
\log_a (xy) = \log_a x + \log_a y
\][/tex]
3. Simplify using logarithm and exponent rules:
\begin{align}
\log_a x^3 & = 3 \log_a x \quad \text{(since [tex]$\log_a x^k = k \log_a x$[/tex])} \\
\log_a \sqrt{x + y} & = \log_a (x + y)^{1/2} = \frac{1}{2} \log_a (x + y) \quad \text{(since [tex]$\log_a x^{1/n} = \frac{1}{n} \log_a x$[/tex])}
\end{align}
4. Putting everything together:
\begin{align}
\log_a \left(\frac{x^3 y}{\sqrt{x + y}}\right) & = \log_a (x^3 y) - \log_a (\sqrt{x + y}) \\
& = (\log_a x^3 + \log_a y) - \left( \frac{1}{2} \log_a (x + y) \right) \\
& = (3 \log_a x + \log_a y) - \left(\frac{1}{2} \log_a (x + y)\right)
\end{align}
Therefore, the expanded form of the given logarithmic expression is:
[tex]\[
\log_a \left(\frac{x^3 y}{\sqrt{x + y}}\right) = 3 \log_a x + \log_a y - \frac{1}{2} \log_a (x + y)
\][/tex]
This step-by-step process uses the properties of logarithms to simplify the expression. Each step adheres to the logarithmic rules given at the beginning.
