Question

Find the exact value of each expression (without a calculator)? `ln(1/e) and log_11sqrt11`

Answer 1

`-1and1/2`

Explanation:

`color(red)((1)log_aa=1rArrlog_ee=1rArrlne=1)`
`color(red)((2)ln1=0)`
`color(red)((3)lnx-lny=ln(x/y))`
`color(red)((4)lnx^n=n*lnx)`
Using (3),(2),and (1) respectively,we get
`ln(1/e)=ln1-lne=0-1=-1`
Using (4) and then (1),we get
`log_11sqrt11=log_11(11)^(1/2)=1/2*log_11 11=1/2*1=1/2`