Question

NO LINKS!!

Use the properties to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume the variable is positive)

1. ln((xy)^5)

2. ln (seventh root(t))

3. ln (√x²/y^5)

Answer 1

`\textsf{1.} \quad 5 \ln x + 5 \ln y`

`\textsf{2.} \quad (1)/(7) \ln t`

`\textsf{3.} \quad \ln x - (5)/(2)\ln y \;\; \;\;\textsf{or} \;\;\;\; \ln x - 5\ln y`

Explanation:

`\boxed{\begin{minipage}{6 cm}\underline{Natural log laws}\\\\Product law: \;$\ln xy=\ln x + \ln y$\\\\Quotient law: $\ln \left((x)/(y)\right) = \ln x - \ln y$\\\\Power law: \;\;\;\;$\ln x^n=n \ln x$\\\end{minipage}}`

Question 1

Apply the power law followed by the product law:

`\begin{aligned}\ln (xy)^5 & = 5 \ln (xy)\\ & = 5\left( \ln x + \ln y \right) \\ & = 5 \ln x + 5 \ln y\end{aligned}`

Question 2

`\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{(1)/(n)}`

then apply the power law:

`\begin{aligned}\ln \sqrt[7]{t} & = \ln t^{(1)/(7)} \\ & = (1)/(7) \ln t\end{aligned}`

Question 3

It is not completely clear where the square root sign begins and ends, so I have provided answers for both permutations:

`\begin{aligned}\ln \left(\sqrt{(x^2)/(y^5)}\right) & = \ln \left((√(x^2))/(√(y^5))\right) \\\\& = \ln \left(\frac{x}{y^{(5)/(2)}}\right) \\\\&=\ln x - \ln y^{(5)/(2)}\\\\&=\ln x - (5)/(2)\ln y\end{aligned}`

`\begin{aligned}\ln \left((√(x^2))/(y^5)\right)& = \ln \left((x)/(y^5)\right) \\\\&=\ln x - \ln y^5\\\\&=\ln x - 5\ln y\end{aligned}`

Answer 2

Properties of log:

• 
  `log\ ab=log\ a + log\ b`
• 
  `log\ a/b =log\ a-log\ b`
• 
  `log\ a^b=b\ log\ a`

Evaluate given using the properties above:

Q1

• 
  `ln((xy)^5) = 5ln(xy) = 5(ln x + ln y) = 5\ ln\ x + 5\ ln\ y`

Q2

• 
  `ln(\sqrt[7]{t} ) = ln(t^(1/7))=1/7\ ln\ t`

Q3

• 
  `ln(√(x^2)/y^5)=ln(x/y^5)= ln\ x - ln\ y^5 = ln\ x - 5\ ln\ y`