Question

Evaluate the following:

`log  81/8 + 2 log  2/3 - 3 log  3/2 + log  3/4`

Answer 1

`log  81/8 + 2 log  2/3 - 3 log  3/2 + log  3/4`

Using the rule n log ⁡a = log ⁡aⁿ,

`log  81/8 + log (2/3)^2 - log (3/2)^3 + log  3/4`

Simplify the powers:

`(2/3)^2 = 4/9`

`(3/2)^3 = 27/8`

= `log  81/8 + log  4/9 - log  27/8 + log  3/4`

Now using combine all logs,

log a + log b − log c + log d = `log ((a xx b xx d)/c)`

= `log((81/8 xx 4/9 xx 3/4)/(27/8))`

Simplify step-by-step:

First, multiply the numerators and denominators:

= `log ((81 xx 4 xx 3)/(8 xx 9 xx 4) xx 8/27)`

Simplify by canceling common factors 8 and 4:

= `log ((81 xx 3)/(9 xx 27))`

= `log (243/243)`

= log (1)

= 0