1/4 log base 2 (4)+ 1/8 log base 2 (8)+ 3/8 log base 2 (8/3)+ 1/4 log base 2 (4)
Assuming base 2 throughout, note that
log4 = 2
log8 = 3
So, we have
1/4 log4 + 1/8 log8 + 3/8 log(8/3) + 1/4 log 4
1/4 (2) + 1/8 (3) + 3/8 (log8-log3) + 1/4 (2)
1/4 (2) + 1/8 (3) + 3/8 (3-log3) + 1/4 (2)
1/4 (2) + 1/8 (3) + 3/8 (3)- 3/8 log3 + 1/4 (2)
1/2 + 3/8 + 9/8 + 1/2 - 3/8 log3
5/2 - 3/8 log3
plz can u give answer in decimal
heck no -- use your own calculator, cantcha?
but, since you don't have a base-2 log button, you'll have to use the change of base formula:
log23 = log3/log2
and you can use whatever base you want for that.
To simplify the expression 1/4 log base 2 (4) + 1/8 log base 2 (8) + 3/8 log base 2 (8/3) + 1/4 log base 2 (4), we can start by applying logarithmic properties.
1) Logarithmic property: log base b (a^c) = c * log base b (a)
Using this property, we can rewrite the expression:
1/4 log base 2 (4) + 1/8 log base 2 (8) + 3/8 log base 2 (8/3) + 1/4 log base 2 (4)
= (1/4)(log base 2 (2^2)) + (1/8)(log base 2 (2^3)) + (3/8)(log base 2 ((8^2)/(3^2))) + (1/4)(log base 2 (2^2))
2) Logarithmic property: log base b (b) = 1
Using this property, we simplify further:
= (1/4)(2) + (1/8)(3) + (3/8)(log base 2 (64/9)) + (1/4)(2)
= 1/2 + 3/8 + (3/8)(log base 2 (64/9)) + 1/2
= 3/4 + (3/8)(log base 2 (64/9))
Now, let's focus on the term (3/8)(log base 2 (64/9)).
3) Logarithmic property: log base b (a * c) = log base b (a) + log base b (c)
Using this property, we can rewrite the term:
(3/8)(log base 2 (64/9))
= (3/8)(log base 2 (64) + log base 2 (1/9))
= (3/8)(log base 2 (2^6) + log base 2 (1/9))
= (3/8)(6 + log base 2 (1/9))
= (3/8)(6 - log base 2 (9))
Finally, we can substitute this back into the previous expression:
= 3/4 + (3/8)(6 - log base 2 (9))
And that would be the simplified form of the given expression.