Question

Help pls!!! With explanations

Answer 1

`(log_(2)(24))/(log_(96)(2)) - (log_(2)(192))/(log_(12)(2))`

log₂(24):
`2^(x) = 24`

`ln(2^(x)) = ln(24)`

`xln(2) = ln(24)`

`x = (ln(24))/(ln(2))`

`x = (ln(8 * 3))/(ln(2))`

`x = (ln(8) + ln(3))/(ln(2))`

`x = (3ln(2) + ln(3))/(ln(2))`

`x = (3ln(2))/(ln(2)) + (ln(3))/(ln(2))`

`x = 3 + 1.1`

`x = 4.1`

log₉₆(2):
`96^(x) = 2`

`ln(96^(x)) = ln(2)`

`xln(96) = ln(2)`

`xln(32 * 3) = ln(2)`

`x(ln(32) + ln(3)) = ln(2)`

`(x(ln(32) + ln(3)))/(ln(32) + ln(3)) = (ln(2))/(ln(32) + ln(3))`

`x = (ln(2))/(ln((2)^(5)) + ln(3))`

`x = (ln(2))/(5ln(2) + ln(3))`

`x = 0.7`


`(log_(2)(24))/(log_(96)(2)) = (3 + (ln(3))/(ln(2)))/((ln(2))/(5ln(2) + ln(3))) = (4.1)/(0.7) = 5(6)/(7)`

log₂(192):
`2^(x) = 192`

`ln(2^(x)) = ln(192)`

`xln(2) = ln(64 * 3)`

`xln(2) = ln(64) + ln(3)`

`xln(2) = ln((2)^(6)) + ln(3)`

`xln(2) = 6ln(2) + ln(3)`

`(xln(2))/(ln(2)) = (6ln(2) + ln(3))/(ln(2))`

`x = (6ln(2))/(ln(2)) + (ln(3))/(ln(2))`

`x = 6 + 1.1`

`x = 7.1`

log₁₂(2):
`12^(x) = 2`

`ln(12^(x)) = ln(2)`

`xln(12) = ln(2)`

`xln(4 * 3) = ln(2)`

`x(ln(4) + ln(3)) = ln(2)`

`(x(ln(4) + ln(3)))/(ln(4) + ln(3)) = (ln(2))/(ln(4) + ln(3))`

`x = (ln(2))/(ln((2)^(2)) + ln(3))`

`x = (ln(2))/(2ln(2) + ln(3))`

`x = 0.3`


`(log_(2)(192))/(log_(12)(2)) = (log_(2)(192))/(log_(12)(2)) = (7.1)/(0.3) = 23(2)/(3)`


`5(6)/(7) - 23(2)/(3)`

`-17(17)/(21)`