Question
How can you condense 1n 11 - 3 1n p
Answers
GPT-4o mini
To condense the expression \( \ln 11 - 3 \ln p \), you can use the properties of logarithms. Here are the steps:
1. **Use the power rule**: The power rule states that \( a \ln b = \ln(b^a) \). Therefore, \( -3 \ln p \) can be written as \( \ln(p^{-3}) \).
So, the expression becomes:
\[
\ln 11 - 3 \ln p = \ln 11 + \ln(p^{-3})
\]
2. **Use the product rule**: The product rule states that \( \ln a + \ln b = \ln(ab) \). Thus, you can combine both logarithmic terms:
\[
\ln 11 + \ln(p^{-3}) = \ln(11 \cdot p^{-3})
\]
Combining these steps results in:
\[
\ln 11 - 3 \ln p = \ln\left(\frac{11}{p^3}\right)
\]
Therefore, the condensed form of \( \ln 11 - 3 \ln p \) is:
\[
\ln\left(\frac{11}{p^3}\right)
\]
1. **Use the power rule**: The power rule states that \( a \ln b = \ln(b^a) \). Therefore, \( -3 \ln p \) can be written as \( \ln(p^{-3}) \).
So, the expression becomes:
\[
\ln 11 - 3 \ln p = \ln 11 + \ln(p^{-3})
\]
2. **Use the product rule**: The product rule states that \( \ln a + \ln b = \ln(ab) \). Thus, you can combine both logarithmic terms:
\[
\ln 11 + \ln(p^{-3}) = \ln(11 \cdot p^{-3})
\]
Combining these steps results in:
\[
\ln 11 - 3 \ln p = \ln\left(\frac{11}{p^3}\right)
\]
Therefore, the condensed form of \( \ln 11 - 3 \ln p \) is:
\[
\ln\left(\frac{11}{p^3}\right)
\]