Respuesta :
In logarithm
multiplication translates to addition on expansion and a fraction or basically division translates to subtraction
so the number question can be rewritten as
[tex]\begin{gathered} \ln (x^{\frac{1}{2}}_{}\times y^3) \\ on\text{ expansion multiplication changes to addition so we have} \\ \ln (x^{\frac{1}{2}})+ln(y^3) \end{gathered}[/tex][tex]\begin{gathered} \text{Next, we bring out the exponents }\frac{1}{2}\text{ on x and 3 on y to the front of the ln} \\ \text{This too is also a law} \end{gathered}[/tex]so, we have
[tex]\frac{1}{2}\ln (x)\text{ + 3ln(y)}[/tex][tex]\begin{gathered} \text{Next combine the }\frac{1}{2\text{ }}and\text{ the ln} \\ so\text{ we have} \\ \frac{\ln(x)}{2}+3\ln (x) \end{gathered}[/tex][tex]\begin{gathered} 2.\text{ log}\sqrt[4]{x^3} \\ \text{Here, first we change }\sqrt[4]{x^3}^{} \\ to\text{ a normal form using laws of indices} \end{gathered}[/tex][tex]\begin{gathered} \text{According to indices }\sqrt[4]{x} \\ ^{}is\text{ the same as} \\ \\ x^{\frac{1}{4}} \end{gathered}[/tex][tex]\begin{gathered} \text{Applying this to the question at hand} \\ \sqrt[4]{x^3} \\ is\text{ the same as} \\ x^{3\times\frac{1}{4}} \\ =x^{\frac{3}{4}} \end{gathered}[/tex]so the question now becomes
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