Question

How do you solve `log_2 (3x)-log_2 7=3`?

Answer 1

Use a property of logs to simplify and solve an algebraic equation to get `x=56/3`.

Explanation:

Begin by simplifying `log_2 3x-log_2 7` using the following property of logs:
`loga-logb=log(a/b)`
Note that this property works with logs of every base, including `2`.

Therefore, `log_2 3x-log_2 7` becomes `log_2 ((3x)/7)`. The problem now reads:
`log_2 ((3x)/7)=3`

We want to get rid of the logarithm, and we do that by raising both sides to the power of `2`:
`log_2 ((3x)/7)=3`
`->2^(log_2 ((3x)/7))=2^3`
`->(3x)/7=8`

Now we just have to solve this equation for `x`:
`(3x)/7=8`
`->3x=56`
`->x=56/3`

Since this fraction cannot be simplified further, it is our final answer.