Use the properties of logarithms to condense the expression into a single logarithm. Show your work by writing on paper or using the mathtype option. ln m+ 5ln p

Answer 1

To condense the expression \( \ln m + 5 \ln p \) into a single logarithm, we will use the properties of logarithms:

1. The property that states \( a \ln b = \ln b^a \).
2. The property that states \( \ln a + \ln b = \ln(ab) \).

Now, let's apply these properties step-by-step:

1. Start with the original expression: \[ \ln m + 5 \ln p \]

2. Apply the first property to \( 5 \ln p \): \[ 5 \ln p = \ln p^5 \]

   So we can rewrite the expression as: \[ \ln m + \ln p^5 \]

3. Now apply the second property to combine the two logarithms: \[ \ln m + \ln p^5 = \ln(m \cdot p^5) \]

Thus, the condensed expression is: \[ \ln(m \cdot p^5) \]