Using the properties logb(1/m)=−logbm

Question

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

ln(11−p3)
l n left parenthesis 11 minus p cubed right parenthesis

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

ln(113p)
l n left parenthesis Start Fraction 11 over 3 p End Fraction right parenthesis

ln(11−3p)

Answer 1

To condense the expression \( \ln(11) - 3\ln(p) \), we can use the properties of logarithms provided:

1. \( \log_b(1/m) = -\log_b(m) \)
2. \( \log_b(m/n) = \log_b(m) - \log_b(n) \)

Using the second property, we can express the subtraction of logarithms in terms of a fraction:

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

Now, applying the second property again:

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

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

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

So, the correct response is:

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