Question

How do you condense `ln x-4 ln(x + 2) + 3 ln x`?

Answer 1

`lnx-4ln(x+2)+3lnx=ln(x/(x+2))^4`

Explanation:

Some of the basic operations in logarithm are `log_ma+log_mb=log_m(ab)`, `log_ma-log_mb=log_m(a/b)` and `nlog_ma=log_m(a^n)`. Further `ln` is special form where `m=e` i.e. base is `e`. Using them

`lnx-4ln(x+2)+3lnx`

= `lnx-ln(x+2)^4+lnx^3`

= `ln((x×x^3)/(x+2)^4)`

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

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