Question

NO LINKS!!
There is a mistake in each of these problems. Please help

Answer 1

`\textsf{ii)} \quad \ln \left((x^3)/(y^2)\right)`

`\textsf{iii)} \quad 20 \log_3 x-4 \log_3y`

Explanation:

Question ii

Given expression:

`5 \ln x-2 \ln (xy)`

`\textsf{Apply the power law}: \quad n \ln x=\ln x^n`

`\implies \ln (x^5) - \ln ((xy)^2)`

`\textsf{Apply the quotient law}: \quad \ln x - \ln y=\ln \left((x)/(y)\right)`

`\implies \ln \left((x^5)/((xy)^2)\right)`

This is where the error occurred in the original calculation as the log quotient law was applied incorrectly.

`\textsf{Apply exponent rule} \quad (ab)^c=a^(c)b^c\quad \textsf{to the denominator}:`

`\implies \ln \left((x^5)/(x^2y^2)\right)`

Simplify the fraction:

`\implies \ln \left((x^3)/(y^2)\right)`

Question iii

Given expression:

`\log_3\left((x^5)/(y)\right)^4`

`\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax`

`\implies 4\log_3\left((x^5)/(y)\right)`

`\textsf{Apply the log quotient law}: \quad n\log_a \left((x)/(y)\right)=n\log_ax - n\log_ay`

`\implies 4 \log_3 x^5-4 \log_3y`

This is where the error occurred in the original calculation as the coefficient 4 was not applied to the both logarithms when applying the log quotient law.

`\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax`

`\implies 5 \cdot 4 \log_3 x-4 \log_3y`

`\implies 20 \log_3 x-4 \log_3y`

Answer 2

ii) Reduce :

`$\begin{aligned} 5 \ln x-2 \ln (x y) & =\ln x^5-\ln (x y)^2 \\ & =\ln \left((x^5)/((x y)^2)\right) \\ & =\ln \left((x^5)/(x^2 y^2)\right) \\ & =\ln \left((x^3)/(y^2)\right)\end{aligned}$`

iii ) Expand :

`\begin{aligned}\log _3\left((x^5)/(y)\right)^4 & =4 \log _3\left((x^5)/(y)\right) \\& =4 \log _3 x^5-4 \log _3 y \\& =20 \log _3 x-4 \log _3 y\end{aligned}`