Question

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45. Express 3log₅x + log₅y - 2log₅w as a single logarithm

46. Find the missing value: log21 -2log7 + log28 = log__

47. Expand ln[(x - 4)(2x+5)²]

Answer 1

`\textsf{45.} \quad \log_5\left((x^3y)/(w^2)\right)`

`\textsf{46.} \quad \log 12`

`\textsf{47.} \quad \ln(x-4)+2\ln(2x+5)`

Explanation:

`\boxed{\begin{minipage}{8cm}\underline{Log laws}\\\\Product law:\quad\:$\log_axy=\log_ax + \log_ay$\\\\Quotient law:\;\;\;$\log_a \left((x)/(y)\right)=\log_ax - \log_ay$\\\\Power law:\quad\;\;\;\:$\log_ax^n=n\log_ax$\\\end{minipage}}`

Question 45

`\begin{aligned}3\log_5x + \log_5y - 2\log_5w&=\log_5x^3 + \log_5y - \log_5w^2\\&=\log_5\left(x^3 \cdot y\right)- \log_5w^2\\&=\log_5\left((x^3y)/(w^2)\right)\end{aligned}`

Question 46

`\begin{aligned}\log21 - 2 \log7 + \log28 &=\log21 - \log7^2 + \log28\\&=\log21 - \log49 + \log28\\&=\log\left((21)/(49)\right) + \log28\\&=\log\left((3)/(7) \cdot 28\right)\\&=\log\left((84)/(7)\right)\\&=\log12\end{aligned}`

Question 47

`\begin{aligned}\ln \left[(x - 4)(2x+5)^2\right]&=\ln(x-4)+\ln(2x+5)^2\\&=\ln(x-4)+2\ln(2x+5)\\\end{aligned}`