Question

Log10base10-2log(1/5)base10-log2.5base10

Answer 1

log 10 - 2 log (1/5) - log 2.5 , (if the base is 10 we don't have to say it)
= 1 - 2(log 1 - log 5) - log (5/2)
= 1 - 2( 0 - log5) - (log5 - log2)
= 1 + 2lo5 - log5 + log2
= 1 + log5 + log2
= 1 + log(5*2)
= 1 + log10
= 1 + 1
= 2

Answer 2

OR

It is not necessary to indicate base 10 because the notation for that logarithm is just log.

You can write:

log ( 10 )₁₀ - 2 log ( 1 / 5 )₁₀ - log ( 2.5 )₁₀

as

log ( 10 ) - 2 log ( 1 / 5 ) - log ( 2.5 )

1 / 5 = 2 / 10

log ( a / b ) = log ( a ) - log ( b )

log ( 1 / 5 ) = log ( 2 / 10 ) = log ( 2 ) - log ( 10 )

2.5 = 10 / 4

log ( 2.5 ) = log ( 10 ) - log ( 4 )

log ( 10 ) - 2 log ( 1 / 5 ) - log ( 2.5 ) =

log ( 10 ) - 2 log ( 2 / 10 ) - log ( 2.5 ) =

log ( 10 ) - 2 [ log ( 2 ) - log ( 10 ) ] - [ log ( 10 ) - log ( 4 ) ] =

log ( 10 ) - 2 log ( 2 ) + 2 log ( 10 ) - log ( 10 ) + log ( 4 ) =

log ( 10 ) - log ( 10 ) - 2 log ( 2 ) + 2 log ( 10 ) + log ( 4 ) =

- 2 log ( 2 ) + 2 log ( 10 ) + log ( 4 ) =

Since:

log ( 10 ) = 1

- 2 log ( 2 ) + 2 log ( 10 ) + log ( 4 ) =

- 2 ∙ log ( 2 ) + 2 ∙ 1 + log ( 4 ) =

- 2 log ( 2 ) + 2 + log ( 4 )

log ( aⁿ ) = n log (a )

Since 4 = 2 ²

log ( 4 ) = log ( 2 ² ) = 2 log ( 2 )

- 2 log ( 2 ) + 2 + log ( 4 ) = - 2 log ( 2 ) + 2 + 2 log ( 2 ) = 2

So:

log ( 10 )₁₀ - 2 log ( 1 / 5 )₁₀ - log ( 2.5 )₁₀ = 2

Answer 3

If the base of a logarithm is 10, just omit the word "base" e.g log 5base 10 is the same as log 5

log 10- 2log (1/5)- log 2.5
2log (1/5)= log (1/5)^2 =log (1/25) since mlog n =log n^m

log 10- log (1/25) -log 2.5
since log a -log b=log (a/b)
then, log 10-log (1/25) = log (10/ 1/25) = log 250

log 250-log 2.5
=log (250/2.5)
=log 100
= 2