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]
