Question
Expand the logarithm. Simplify completely. You must show thorough, handwritten work. You may only use scratch paper, a writing utensil, and a calculator that does not have a computer algebra system.
log(x2y100)
log(x2y100)
Answers
GPT-4o mini
To expand and simplify the expression \( \log(x^2y^{100}) \), we will use the properties of logarithms. Here are the steps in detail:
1. **Use the Product Rule of Logarithms**: The product rule states that \( \log(ab) = \log(a) + \log(b) \). We can apply this rule since we have a product \( x^2 \) and \( y^{100} \):
\[
\log(x^2y^{100}) = \log(x^2) + \log(y^{100})
\]
2. **Use the Power Rule of Logarithms**: The power rule states that \( \log(a^b) = b \log(a) \). We can apply this rule to both terms:
- For \( \log(x^2) \):
\[
\log(x^2) = 2 \log(x)
\]
- For \( \log(y^{100}) \):
\[
\log(y^{100}) = 100 \log(y)
\]
3. **Combine the results**: Now we can write the expanded form by substituting back into our equation:
\[
\log(x^2y^{100}) = 2 \log(x) + 100 \log(y)
\]
Thus, the final simplified expansion of \( \log(x^2y^{100}) \) is:
\[
\boxed{2 \log(x) + 100 \log(y)}
\]
1. **Use the Product Rule of Logarithms**: The product rule states that \( \log(ab) = \log(a) + \log(b) \). We can apply this rule since we have a product \( x^2 \) and \( y^{100} \):
\[
\log(x^2y^{100}) = \log(x^2) + \log(y^{100})
\]
2. **Use the Power Rule of Logarithms**: The power rule states that \( \log(a^b) = b \log(a) \). We can apply this rule to both terms:
- For \( \log(x^2) \):
\[
\log(x^2) = 2 \log(x)
\]
- For \( \log(y^{100}) \):
\[
\log(y^{100}) = 100 \log(y)
\]
3. **Combine the results**: Now we can write the expanded form by substituting back into our equation:
\[
\log(x^2y^{100}) = 2 \log(x) + 100 \log(y)
\]
Thus, the final simplified expansion of \( \log(x^2y^{100}) \) is:
\[
\boxed{2 \log(x) + 100 \log(y)}
\]