Question

Expand using the properties and rules for logarithms

Answer 1

Consider expression
`\log_{(1)/(2)}\left((3x^2)/(2)\right).`

1. Use property

`\log_a(b)/(c)=\log_ab-\log_ac.`

Then

`\log_{(1)/(2)}\left((3x^2)/(2)\right)=\log_{(1)/(2)}3x^2-\log_{(1)/(2)}2.`

2. Use property

`\log_abc=\log_ab+\log_ac.`

Then

`\log_{(1)/(2)}\left((3x^2)/(2)\right)=\log_{(1)/(2)}3x^2-\log_{(1)/(2)}2=\log_{(1)/(2)}3+\log_{(1)/(2)}x^2-\log_{(1)/(2)}2.`

3. Use property

`\log_ab^k=k\log_ab.`

Then

`\log_{(1)/(2)}\left((3x^2)/(2)\right)=\log_{(1)/(2)}3+\log_{(1)/(2)}x^2-\log_{(1)/(2)}2=\log_{(1)/(2)}3+2\log_{(1)/(2)}x-\log_{(1)/(2)}2.`

4. Use property

`\log_(a^k)b=(1)/(k)\log_ab.`

Then

`\log_{(1)/(2)}\left((3x^2)/(2)\right)=\log_{(1)/(2)}3+2\log_{(1)/(2)}x-\log_{(1)/(2)}2=\log_{(1)/(2)}3+2\log_{(1)/(2)}x-\log_(2^(-1))2=\\ \\=\log_{(1)/(2)}3+2\log_{(1)/(2)}x+\log_22=\log_{(1)/(2)}3+2\log_{(1)/(2)}x+1.`

Answer: correct option is B.