Question

How do you expand `log_bsqrt(57/74)`?

Answer 1

`1/2log_b57-1/2log_b74`

Explanation:

There are certain rules to logratithims. You can find the complete list here, but the one that applies here is the second rule:

`logb(m/n) = logb(m) – logb(n)`

Using this law, we can solve `logbsqrt(57/74)`:
`logb(sqrt(57))/(sqrt(74))`
`logbsqrt(57)-logbsqrt(74)`
We can stop here, but I'm going to keep going and expand it as much as I can
`log_b57^.5-log_b74^.5`
`.5log_b57-.5log_b74`
`1/2log_b57-1/2log_b74`

That's as expanded as it gets! Good work

Answer 2

`log_b(sqrt(57/74))=1/2log_b57-1/2log_b74`

Explanation:

`log_b(sqrt(57/74))=log_b((57/74)^(1/2))`

Use the rule `loga^b=bloga`

`log_b((57/74)^(1/2))=1/2log_b(57/74)`

Use the rule `log(a/b)=loga-logb`, and don't forget that the `1/2` belongs to the whole expression.

`1/2log_b(57/74)=1/2log_b57-1/2log_b74`