Use the properties of logarithms to write the following expression as a single logarithm:

[tex]\[ \ln m + 9 \ln p - 9 \ln m \][/tex]

Provide your answer below:



Answer :

Certainly! Let's simplify the given expression step-by-step using the properties of logarithms.

Starting expression:
[tex]\[ \ln(m) + 9\ln(p) - 9\ln(m) \][/tex]

First, combine the logarithms involving [tex]\( \ln(m) \)[/tex]:
[tex]\[ \ln(m) - 9\ln(m) = (1 - 9)\ln(m) = -8\ln(m) \][/tex]

Now, the expression becomes:
[tex]\[ -8\ln(m) + 9\ln(p) \][/tex]

Next, we use the property of logarithms [tex]\( a\ln(b) = \ln(b^a) \)[/tex] to rewrite the terms:
[tex]\[ -8\ln(m) = \ln(m^{-8}) \][/tex]
[tex]\[ 9\ln(p) = \ln(p^9) \][/tex]

The expression now looks like this:
[tex]\[ \ln(m^{-8}) + \ln(p^9) \][/tex]

Finally, apply the property [tex]\( \ln(b) + \ln(c) = \ln(bc) \)[/tex] to combine the logarithms into a single logarithm:
[tex]\[ \ln(m^{-8}) + \ln(p^9) = \ln(m^{-8} \cdot p^9) \][/tex]

Therefore, the expression simplified into a single logarithm is:
[tex]\[ \ln(m^{-8} \cdot p^9) \][/tex]