Question
Use the properties of logarithms to condense the expression into a single logarithm. Show your work by writing on paper or using the mathtype (Σ) option.
ln m+ 5ln p
ln m+ 5ln p
Answers
Answered by
GPT-4o mini
To condense the expression \( \ln m + 5 \ln p \) into a single logarithm, we can use the properties of logarithms. Specifically, we will use the following properties:
1. The property of logarithm for multiplication: \( \ln a + \ln b = \ln(ab) \).
2. The property of logarithm for powers: \( k \ln a = \ln(a^k) \).
Now, let's apply these properties step by step:
1. Start with the expression:
\[
\ln m + 5 \ln p
\]
2. Use the power property on \( 5 \ln p \):
\[
5 \ln p = \ln(p^5)
\]
3. Substitute back into the original expression:
\[
\ln m + \ln(p^5)
\]
4. Now apply the multiplication property:
\[
\ln m + \ln(p^5) = \ln(m \cdot p^5)
\]
So, the condensed form of the expression \( \ln m + 5 \ln p \) is:
\[
\ln(m \cdot p^5)
\]
1. The property of logarithm for multiplication: \( \ln a + \ln b = \ln(ab) \).
2. The property of logarithm for powers: \( k \ln a = \ln(a^k) \).
Now, let's apply these properties step by step:
1. Start with the expression:
\[
\ln m + 5 \ln p
\]
2. Use the power property on \( 5 \ln p \):
\[
5 \ln p = \ln(p^5)
\]
3. Substitute back into the original expression:
\[
\ln m + \ln(p^5)
\]
4. Now apply the multiplication property:
\[
\ln m + \ln(p^5) = \ln(m \cdot p^5)
\]
So, the condensed form of the expression \( \ln m + 5 \ln p \) is:
\[
\ln(m \cdot p^5)
\]