Express each logarithm in terms of In 2 and In 5. In(32)/(125) In12.5

Question

Express each logarithm in terms of In 2 and In 5.


`In(32)/(125)`


`In12.5`

Answer 1

`5ln2-3ln5`

`2ln5-ln2`

Explanation:

∵
`32=2^(5)`

∵
`125=5^(3)`

∵
`ln(32)/(125)=ln(2^(5))/(5^(3))`

∴
`ln2^(5)-ln5^(3)=5ln2-3ln5`

∵
`12.5=(125)/(10)=(25)/(2)=(5^(2))/(2)`

∴
`ln12.5=ln(5^(2))/(2)`

∴
`ln5^(2)-ln2=2ln5-ln2`