Use the properties of logarithms to expand the logarithm as much as possible. Rewrite the expression as
a sum, difference, or product of logs.

Ask by Mccoy Page. in the United States

Jan 24,2025

Question

Use the properties of logarithms to expand the logarithm as much as possible. Rewrite the expression as a sum, difference, or product of logs. $$ \log \left(\sqrt{x^{5} y^{-6}}\right) $$

UpStudy AI · Accepted Answer

To expand the logarithmic expression $$\log \left(\sqrt{x^{5} y^{-6}}\right)$$, we'll use the properties of logarithms step by step.

1. **Rewrite the Square Root as an Exponent:**
   $$
   \sqrt{x^{5} y^{-6}} = (x^{5} y^{-6})^{1/2}
   $$

2. **Apply the Logarithm Power Rule ($$\log(a^k) = k \log(a)$$):**
   $$
   \log \left((x^{5} y^{-6})^{1/2}\right) = \frac{1}{2} \log(x^{5} y^{-6})
   $$

3. **Expand the Logarithm of a Product ($$\log(ab) = \log(a) + \log(b)$$):**
   $$
   \frac{1}{2} \log(x^{5} y^{-6}) = \frac{1}{2} \log(x^{5}) + \frac{1}{2} \log(y^{-6})
   $$

4. **Apply the Power Rule Again to Each Term:**
   $$
   \frac{1}{2} \log(x^{5}) = \frac{5}{2} \log(x)
   $$
   $$
   \frac{1}{2} \log(y^{-6}) = \frac{-6}{2} \log(y) = -3 \log(y)
   $$

5. **Combine the Terms:**
   $$
   \frac{5}{2} \log(x) - 3 \log(y)
   $$

**Final Expanded Form:**
$$
\frac{5}{2} \log x\ -\ 3\,\log y
$$
$$ \frac{5}{2} \log x - 3 \log y $$

Use the properties of logarithms to expand the logarithm as much as possible. Rewrite the expression as
a sum, difference, or product of logs.

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Pre Algebra Questions