Using the properties logb(1/m)=−logbm

and logb(m/n)=logbm−logbn
, how can ln11−3lnp
be condensed?(1 point)
Responses

ln(11/3p)
l n left parenthesis Start Fraction 11 over 3 p End Fraction right parenthesis

ln(11−3p)
l n left parenthesis 11 minus 3 p right parenthesis

ln(11/p3)
l n left parenthesis Start Fraction 11 over p cubed End Fraction right parenthesis

ln(11−p3)

1 answer

To condense the expression \( \ln 11 - 3 \ln p \) using the properties of logarithms, we can apply the second property mentioned, which is:

\[ \log_b(m/n) = \log_b(m) - \log_b(n). \]

Additionally, we can use the first property that allows us to express \( k \log_b(m) \) as \( \log_b(m^k) \).

Starting with the expression:

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

we can use the property \( k \log_b(m) = \log_b(m^k) \) on \( -3 \ln p \):

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

Now, applying the property of logarithms about the difference, we can combine the two logarithms:

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

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

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

So the correct response is:

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