Find the value of log✓15+log2-log6-½
that would be
1/2 log15 + log2 + 1/2 log6
log √(15*4*6) = log√360 = 1/2 log360 = ____
If log means log₁₀ ( logarithm for base 10 ) then:
log ( √10 ) = 1 / 2 log ( 10 ) = 1 / 2 ∙ 1 = 1 / 2
1 / 2 = log ( √10 )
log ( √15 ) + log ( 2 ) - log ( 6 ) - 1 / 2 =
log ( √15 ) + log ( 2 ) - log ( 6 ) - log ( √10 ) =
1 / 2 log ( 15 ) + log ( 2 ) - log ( 6 ) - 1 / 2 log ( 10 ) =
1 / 2 log ( 3 ∙ 5 ) + log ( 2 ) - log ( 2 ∙ 3 ) - 1 / 2 log ( 2 ∙ 5 ) =
1 / 2 [ log ( 3 ) + log ( 5 ) ] + log ( 2 ) - [ log ( 2 ) + log ( 3 ) ] - 1 / 2 [ log ( 2 ) + log ( 5 ) ] =
1 / 2 log ( 3 ) + 1 / 2 log ( 5 ) + log ( 2 ) - log ( 2 ) - log ( 3 ) - 1 / 2 log ( 2 ) - 1 / 2 log ( 5 ) =
1 / 2 log ( 3 ) + 1 / 2 log ( 5 ) - log ( 3 ) - 1 / 2 log ( 2 ) - 1 / 2 log ( 5 ) =
1 / 2 log ( 3 ) - log ( 3 ) + 1 / 2 log ( 5 ) - 1 / 2 log ( 5 ) - 1 / 2 log ( 2 ) =
- 1 / 2 log ( 3 ) - 1 / 2 log ( 2 ) =
- 1 / 2 [ log ( 3 ) + log ( 2 ) ] =
- 1 / 2 log ( 3 ∙ 2 ) =
- 1 / 2 log ( 6 ) = log ( 1 / √ 6 )
To find the value of the expression log√15 + log2 - log6 - 1/2, we can simplify each term step by step.
1. Simplify log√15:
The square root of 15 can be written as √(3 * 5).
Using the logarithmic property log(ab) = log(a) + log(b), we can rewrite log√15 as (1/2)log(3 * 5).
2. Simplify log2:
This term represents the logarithm of 2 to the base 10, which is commonly denoted as log(2).
Since log(2) is a constant, we keep it as it is.
3. Simplify log6:
This term represents the logarithm of 6 to the base 10, which is commonly denoted as log(6).
Since log(6) is a constant, we keep it as it is.
Now, let's put all these terms together:
log√15 + log2 - log6 - 1/2
(1/2)log(3 * 5) + log(2) - log(6) - 1/2
Now, we can apply mathematical rules to simplify further:
(1/2)(log(3) + log(5)) + log(2) - log(6) - 1/2
Using the logarithmic property log(ab) = log(a) + log(b):
(1/2)(log(3) + log(5)) + log(2) - log(6) - 1/2
Now, we can use logarithmic identity to combine the terms:
(1/2)log(3) + (1/2)log(5) + log(2) - log(6) - 1/2
We have simplified the expression as much as possible. However, without specific values for log(3), log(5), log(2), and log(6), we cannot calculate the exact numerical value.