Question

using the properties logb(1m)=−logbm and logb(mn)=logbm−logbn , how can ln11−3lnp be condensed?

Answer 1

To condense the expression \( \ln 11 - 3 \ln p \), we can use the properties of logarithms. Specifically, we can utilize the property that states \( a \ln b = \ln(b^a) \) and the property that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \).

Starting with the expression:

\[
\ln 11 - 3 \ln p
\]

We can first rewrite \( 3 \ln p \) using the property \( a \ln b = \ln(b^a) \):

\[
3 \ln p = \ln(p^3)
\]

So we replace \( 3 \ln p \) in the original expression:

\[
\ln 11 - 3 \ln p = \ln 11 - \ln(p^3)
\]

Now, we can apply the property \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \):

\[
\ln 11 - \ln(p^3) = \ln \left( \frac{11}{p^3} \right)
\]

Thus, the condensed form of \( \ln 11 - 3 \ln p \) is:

\[
\ln \left( \frac{11}{p^3} \right)
\]