Question
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
, and .

Answer Attempt 1 out of 2

Ask by May Davison. in the United States

Dec 22,2024

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Question Expand the logarithm fully using the properties of logs. Express the final answer in terms of $$ \log x $$, and $$ \log y $$. $$ \log x y^{5} $$ Answer Attempt 1 out of 2

UpStudy AI · Accepted Answer

To fully expand the logarithm $$ \log (x y^{5}) $$ using the properties of logarithms, follow these steps:

### **Step 1: Apply the Product Rule**

The **Product Rule** for logarithms states that:
$$
\log(ab) = \log a + \log b
$$
Apply this rule to $$ \log (x y^{5}) $$:
$$
\log (x y^{5}) = \log x + \log y^{5}
$$

### **Step 2: Apply the Power Rule**

The **Power Rule** for logarithms states that:
$$
\log(a^{k}) = k \log a
$$
Apply this rule to $$ \log y^{5} $$:
$$
\log y^{5} = 5 \log y
$$

### **Step 3: Combine the Results**

Substitute the result from the Power Rule back into the expression from the Product Rule:
$$
\log (x y^{5}) = \log x + 5 \log y
$$

### **Final Expanded Form**

$$
\boxed{ \log x + 5\,\log y }
$$

This is the fully expanded form of $$ \log (x y^{5}) $$ expressed in terms of $$ \log x $$ and $$ \log y $$.
$$ \log x + 5\,\log y $$

Question
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
, and .

Answer Attempt 1 out of 2

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Question Expand the logarithm fully using the properties of logs. Express the final answer in terms of , and . Answer Attempt 1 out of 2