1/3log125/8-2log2/5+log80/125​

1

Step-by-step explanation:

⅓Log₁₀(125/8) – 2Log₁₀(2/5) + Log₁₀(80/125)

The above expression can be simplified as follow:

⅓Log(125/8) – 2Log(2/5) + Log(80/125)

Recall:

a LogM = Log Mᵃ

Therefore,

⅓Log(125/8) – 2Log(2/5) + Log(80/125)

= Log(125/8)¹/³ – Log(2/5)² + Log(80/125)

= Log(5/2) – Log(4/25) + Log(16/25)

Recall:

Log M – Log N = Log M/N

Log(5/2) – Log(4/25) = Log (5/2 ÷ 4/25)

Log(5/2) – Log(4/25) = Log (5/2 × 25/4)

Log(5/2) – Log(4/25) = Log (125/8)

Therefore,

Log(5/2) – Log(4/25) + Log(16/25)

= Log (125/8) + Log(16/25)

Recall:

Log M + Log N = Log MN

Log (125/8) + Log(16/25)

= Log (125/8 × 16/25)

= Log 10

= Log₁₀10 = 1

Therefore,

⅓Log(125/8) – 2Log(2/5) + Log(80/125)

= 1