Question

If `log_b x = 2/3log_b 27 + 2 log_b 2 -log_b 3`, what is `x`?

Answer 1

`x=color(green)(12)`

Explanation:

Things you need to know and remember:
`color(white)("XXX")log_b (p^q) = q * log_b p`
and
`color(white)("XXX")log_b (p) + log_b (q) = log_b (pq)`

`color(red)(2/3log_b 27)`
`color(white)("XXX")=2/3 log_b 3^3 = 2/3 * 3 log_b 3 = color(red)(2 log_b 3)`

`color(blue)(2log_b 2)`
`color(white)("XXX")=log_b 2^2 = color(blue)(log_b 4)`

`color(red)(2/3 log_b 27) + color(blue)(2 log_b 4) - color(black)(log_b 3)`
`color(white)("XXX")=color(red)(2log_b 3) +color(blue)(log_b 4) -color(black)(log_b 3)`

`color(white)("XXX")=log_b 3 + log_b 4`

`color(white)("XXX")=log_b 12`

Therefore
`color(white)("XXX")log_b x = 2/3log_b 27 + 2 log_b 2 -log_b 3`

`color(white)("XXX")rArr log_b x = log_b 12`

`color(white)("XXX")rArr x =12`