Log10base10-2log(1/5)base10-log2.5base10
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
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
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
To solve this expression, let's break it down step by step:
1. log10(base 10) - 2log(1/5)(base 10) - log(2.5)(base 10)
First, let's simplify the second term:
2. log(1/5) (base 10) = log(1) - log(5) (base 10)
Since log(1) equals 0, we have:
= 0 - log(5) (base 10)
= -log(5) (base 10)
Now, let's substitute this value back into the original expression:
3. log10 (base 10) - 2(-log(5) (base 10)) - log(2.5) (base 10)
Next, simplify the third term:
4. log(2.5) (base 10)
Now, let's put all these terms together:
5. log10 (base 10) - 2(-log(5) (base 10)) - log(2.5) (base 10)
Now, we'll apply the logarithmic property log(a) - log(b) = log(a/b), which simplifies the expression further:
6. log10 (base 10) - log(5^2) (base 10) - log(2.5) (base 10)
Since 5^2 equals 25, we have:
7. log10 (base 10) - log(25) (base 10) - log(2.5) (base 10)
Now, use the logarithmic property log(a) - log(b) = log(a/b) again:
8. log10 (base 10) - log(25/2.5) (base 10)
Since 25/2.5 equals 10, we have:
9. log10 (base 10) - log(10) (base 10)
Finally, apply the logarithmic property log(a) (base a) = 1:
10. 1 - 1 = 0
Therefore, the value of the expression log10(base 10) - 2log(1/5)(base 10) - log(2.5)(base 10) is 0.