Using the properties logb(1m)=−logbm

Question

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

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

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

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

ln(11−p3)

Answer 1

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

1. The first property states that \( \log_b(mn) = \log_b(m) + \log_b(n) \), which helps in combining logs when you are adding.

2. The second property states that \( \log_b(m^n) = n \log_b(m) \), which allows us to bring the coefficient of \( \ln(p) \) down as an exponent.

Using these properties:

Starting with the expression: \[ \ln(11) - 3\ln(p) \]

We can apply the second property: \[ 3\ln(p) = \ln(p^3) \]

So, the expression now becomes: \[ \ln(11) - \ln(p^3) \]

Now, we can use the first property to combine the logarithms: \[ \ln(11) - \ln(p^3) = \ln\left(\frac{11}{p^3}\right) \]

Thus, the condensed form of \( \ln(11) - 3\ln(p) \) is: \[ \ln\left(\frac{11}{p^3}\right) \]

Looking at your provided response options, the correct option is: \( \ln\left(\frac{11}{p^3}\right) \).