Question

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

ln z(z − 1)^8, z > 1

Answer 1

• ln(ab)=lna+lnb

`\\ \rm\Rrightarrow lnz(z-1)^8`

`\\ \rm\Rrightarrow lnz+ln(z-1)^8`

• lna^b=blna

`\\ \rm\Rrightarrow lnz+8ln(z-1)`

Answer 2

`\ln z + 8\ln (z-1)`

Explanation:

`\ln z(z-1)^8`

Using the product rule:
`\ln(xy)=\ln x + \ln y`

`\implies \ln z + \ln (z-1)^8`

Using the power rule:
`\ln x^n=n \ln x`

`\implies \ln z + 8\ln (z-1)`