meredith48034
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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 :

mhanifa

Properties of log:

  • [tex]log\ ab=log\ a + log\ b[/tex]
  • [tex]log\ a/b =log\ a-log\ b[/tex]
  • [tex]log\ a^b=b\ log\ a[/tex]

Evaluate given using the properties above:

Q1

  • [tex]ln((xy)^5) = 5ln(xy) = 5(ln x + ln y) = 5\ ln\ x + 5\ ln\ y[/tex]

Q2

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

Q3

  • [tex]ln(\sqrt{x^2}/y^5)=ln(x/y^5)= ln\ x - ln\ y^5 = ln\ x - 5\ ln\ y[/tex]
semsee45

Answer:

[tex]\textsf{1.} \quad 5 \ln x + 5 \ln y[/tex]

[tex]\textsf{2.} \quad \dfrac{1}{7} \ln t[/tex]

[tex]\textsf{3.} \quad \ln x - \dfrac{5}{2}\ln y \;\; \;\;\textsf{or} \;\;\;\; \ln x - 5\ln y[/tex]

Step-by-step explanation:

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

Question 1

Apply the power law followed by the product law:

[tex]\begin{aligned}\ln (xy)^5 & = 5 \ln (xy)\\ & = 5\left( \ln x + \ln y \right) \\ & = 5 \ln x + 5 \ln y\end{aligned}[/tex]

Question 2

[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}[/tex]

then apply the power law:

[tex]\begin{aligned}\ln \sqrt[7]{t} & = \ln t^{\frac{1}{7}} \\ & = \dfrac{1}{7} \ln t\end{aligned}[/tex]

Question 3

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

[tex]\begin{aligned}\ln \left(\sqrt{\dfrac{x^2}{y^5}}\right) & = \ln \left(\dfrac{\sqrt{x^2}}{\sqrt{y^5}}\right) \\\\& = \ln \left(\dfrac{x}{y^{\frac{5}{2}}}\right) \\\\&=\ln x - \ln y^{\frac{5}{2}}\\\\&=\ln x - \dfrac{5}{2}\ln y\end{aligned}[/tex]

[tex]\begin{aligned}\ln \left(\dfrac{\sqrt{x^2}}{y^5}\right)& = \ln \left(\dfrac{x}{y^5}\right) \\\\&=\ln x - \ln y^5\\\\&=\ln x - 5\ln y\end{aligned}[/tex]

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