Answered

If [tex]\(a, x\)[/tex], and [tex]\(y\)[/tex] are positive real numbers, [tex]\(a \neq 1\)[/tex], then:

1. [tex]\(\log_a (xy) = \log_a x + \log_a y\)[/tex]
2. [tex]\(\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y\)[/tex]
3. [tex]\(\log_a x^k = k \log_a x\)[/tex]
4. [tex]\(\log_a a = 1\)[/tex]
5. [tex]\(\log_a 1 = 0\)[/tex]
6. [tex]\(a^{\log_a x} = x\)[/tex]
7. [tex]\(\log_a x = \log_a y\)[/tex] if and only if [tex]\(x = y\)[/tex]
8. [tex]\(\log_{a^k} x = \frac{1}{k} \log_a x\)[/tex]

Example 6.16:

Expand the following



Answer :

GinnyAnswer
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.