2 log5 50-log520-2log510 +log5100

Let me write "log to the base 5 of 50" as Log(5)[50], so,

2Log(5)[50] -Log(5)[20] -2Log(5)[10] +Log(5)[100]

= Log(5)[50^2] -Log(5)[20] -Log(5)[10^2] +Log(5)[100]

= Log(5)[2500] -Log(5)[20] -Log(5)[100] +Log(5)[100]

= Log(5)[2500] -Log(5)[20]

= Log(5)[2500/20]

= Log(5)[125] -------------------the single log

= Log(5)[5*5*5]

= Log(5)[5^3]

= 3Log(5)[5]

= 3*1

= 3 ------------------answer.

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I forgot the Log(5)[40].

My calculator cannot give directly antilogs that are not based on 10 or on "e", so I need to convert first the Log(5)[40] into based on 10 or based on "e".

Conversion:

Log(b)[a] = Log(c)[a] / Log(c)[b]

In base 10,

Log(5)[40]

= Log(10)[40] / Log(10)[5]

= 1.60206 / 0.69897

= 2.292

Or, in base "e" or natural log,

Log(5)[40]

= Ln[40] / Ln[5]

= 3.688879 / 1.609438

= 2.292 --------------same as in base 10.

Therefore, Log(5)[40] = 2.292 ------------answer.