First of all let's recal some properties of logarithmic functions:
[tex]\begin{gathered} (1)\text{ }\log _a(b\cdot c)=\log _ab+\log _ac \\ (2)\text{ }\log _a(\frac{b}{c})=\log _ab-\log _ac \\ (3)\log _a(b^c)=c\cdot\log _ab \end{gathered}[/tex]So now that we have this properties in mind let's take a look at each of the three statements given.
First we have:
[tex]\log _3(cd^4)[/tex]Here we can use property (1) since we have a multiplication inside the logarithm:
[tex]\log _3(cd^4)=\log _3\lbrack(c)\cdot(d^4)\rbrack=\log _3(c)+\log _3(d^4)[/tex]Then we can use property (3) in the second term:
[tex]\log _3(cd^4)=\log _3(c)+\log _3(d^4)=\log _3(c)+4\log _3(d)[/tex]But this last expression is different than the one in the statement:
[tex]\log _3(c)+4\log _3(d)\ne4\log _3(c)+4\log _3(d)[/tex]Then the first statement is False.
In the second statement we have:
[tex]\frac{3}{4}(\ln a+\ln b)=\ln (\sqrt[4]{a^3b^3})[/tex]Let's take the expression inside parenthesis at the left side and use property (1):
[tex]\frac{3}{4}(\ln a+\ln b)=\frac{3}{4}\ln (ab)[/tex]We can use property (3) in this last expression:
[tex]\frac{3}{4}\ln (ab)=\ln \lbrack(ab)^{\frac{3}{4}}\rbrack[/tex]Here is important to recal some properties of powers:
[tex]\begin{gathered} (i)\text{ }A^{\frac{B}{C}}=A^{B\cdot\frac{1}{C}}=(A^B)^{\frac{1}{C}} \\ (ii)\text{ }A^{\frac{1}{B}}=\sqrt[B]{A} \end{gathered}[/tex]So if we use property (i):
[tex]\ln \lbrack(ab)^{\frac{3}{4}}\rbrack=\ln \lbrack(a^3b^3)^{\frac{1}{4}}\rbrack[/tex]And using property (ii) we get:
[tex]\ln \lbrack(a^3b^3)^{\frac{1}{4}}\rbrack)=\ln \sqrt[4]{a^3b^3}[/tex]Which means that:
[tex]\frac{3}{4}(\ln a+\ln b)=\ln \sqrt[4]{a^3b^3}[/tex]Then the second statement is True.
The third statement is:
[tex]3\ln e-2\ln f=\ln (\frac{e^3}{f^3})[/tex]Let's take the expression in the left and use property (3) in both terms:
[tex]3\ln e-2\ln f=\ln e^3-\ln f^2[/tex]Now we use property (2):
[tex]3\ln e-2\ln f=\ln e^3-\ln f^2=\ln (\frac{e^3}{f^2})[/tex]Which proves that the third statement is True.
