Question

Please help with 15, 17 and 19

Answer 1

Given:

15.
`\log_{(1)/(2)}\left((1)/(2)\right)`

17.
`\log_{(3)/(4)}\left((4)/(3)\right)`

19.
`2^(\log_2100)`

To find:

The values of the given logarithms by using the properties of logarithms.

Solution:

15. We have,

`\log_{(1)/(2)}\left((1)/(2)\right)`

Using property of logarithms, we get

`\log_{(1)/(2)}\left((1)/(2)\right)=1`
`[\because \log_aa=1]`

Therefore, the value of
`\log_{(1)/(2)}\left((1)/(2)\right)` is 1.

17. We have,

`\log_{(3)/(4)}\left((4)/(3)\right)`

Using properties of logarithms, we get

`\log_{(3)/(4)}\left((4)/(3)\right)=-\log_{(3)/(4)}\left((3)/(4)\right)`
`[\because \log_a(m)/(n)=-\log_a(n)/(m)]`

`\log_{(3)/(4)}\left((4)/(3)\right)=-1`
`[\because \log_aa=1]`

Therefore, the value of
`\log_{(3)/(4)}\left((4)/(3)\right)` is -1.

19. We have,

`2^(\log_2100)`

Using property of logarithms, we get

`2^(\log_2100)=100`
`[\because a^(\log_ax)=x]`

Therefore, the value of
`2^(\log_2100)` is 100.