Question

I need help expanding some log

Answer 1

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

`\begin{gathered} \ln (x^{(1)/(2)}_{}* y^3) \\ on\text{ expansion multiplication changes to addition so we have} \\ \ln (x^{(1)/(2)})+ln(y^3) \end{gathered}`

`\begin{gathered} \text{Next, we bring out the exponents }(1)/(2)\text{ on x and 3 on y to the front of the ln} \\ \text{This too is also a law} \end{gathered}`

so, we have

`(1)/(2)\ln (x)\text{ + 3ln(y)}`
`\begin{gathered} \text{Next combine the }\frac{1}{2\text{ }}and\text{ the ln} \\ so\text{ we have} \\ (\ln(x))/(2)+3\ln (x) \end{gathered}`
`\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}`
`\begin{gathered} \text{According to indices }\sqrt[4]{x} \\ ^{}is\text{ the same as} \\ \\ x^{(1)/(4)} \end{gathered}`
`\begin{gathered} \text{Applying this to the question at hand} \\ \sqrt[4]{x^3} \\ is\text{ the same as} \\ x^{3*(1)/(4)} \\ =x^{(3)/(4)} \end{gathered}`

so the question now becomes