1/3log125/8-2log2/5+log80/125 (1)/(3) log( (125)/(8) ) - 2 log( (2)/(5) ) + log( (80)/(125) ) ​

Question

1/3log125/8-2log2/5+log80/125


`(1)/(3) log( (125)/(8) ) - 2 log( (2)/(5) ) + log( (80)/(125) )`
​

Answer 1

`3log2`

Explanation:

Given

`(1)/(3) log( (125)/(8) ) - 2 log( (2)/(5) ) + log( (80)/(125) )`

Required

Solve

Apply logarithm rules

`log( \frac{125^{(1)/(3)}}{8^{(1)/(3)}} )- log( (2^2)/(5^2) ) + log( (80)/(125) )`

`log( (5)/(2)} )- log( (4)/(25) ) + log( (80)/(125) )`

Apply quotient rule

`log( (5)/(2) / (4)/(25) ) + log( (80)/(125) )`

`log( (5)/(2) * (25)/(4) ) + log( (80)/(125) )`

`log( (100)/(8) ) + log( (80)/(125) )`

Apply product rule

`log( (100*80)/(8*125) )`

`log( (8000)/(1000) )`

`log(8)`

Express 8 as 2^3

`log(2^3)`

`3log2`