How to simplify?
log(1/2)2 + log(1/2)2^(1/2)
2 = (1/2)^-1, so
log(1/2)2 = -1
log(1/2)2 + log(1/2)2^(1/2)
log(1/2)2 + (1/2)*log(1/2)2
(3/2)*log(1/2)2
(3/2)(-1)
-3/2
How did you get the 3/2?
well, there's
1* log + 1/2 * log
Looks like 3/2 * log
to me...
Do you have trouble with
x + 1/2 x = 3/2 x?
Thanks!
To simplify the expression, let's apply the properties of logarithms.
1. Recall that the logarithm of the base raised to an exponent is equal to the exponent. That is, log base a of a^b = b. Moreover, the logarithm of a product is equal to the sum of the logarithms. That is, log base a of (b * c) = log base a of b + log base a of c.
2. Apply these properties to simplify the given expression:
log(1/2)2 + log(1/2)2^(1/2)
Using the properties, we can rewrite the expression as:
= log(1/2)(2) + log(1/2)(2^(1/2))
3. Now, let's use the property mentioned above. The logarithm base 1/2 of 2 is the exponent to which 1/2 must be raised to obtain 2. Similarly, the logarithm base 1/2 of 2^(1/2) is the exponent to which 1/2 must be raised to obtain 2^(1/2).
Therefore, we can rewrite the expression as:
= log(base 1/2)2 + log(base 1/2)2^(1/2)
4. Now, observe that if 1/2 is raised to some exponent to obtain 2, then the exponent is equal to -1. In other words, (1/2)^(-1) = 2. Similarly, if 1/2 is raised to some exponent to obtain 2^(1/2), then the exponent is equal to -1/2. In other words, (1/2)^(-1/2) = 2^(1/2).
Therefore, the expression can be simplified further as:
= -1 + (-1/2)
5. Finally, combine the terms:
= -1 - 1/2
= -3/2
So, the simplified value of the expression log(1/2)2 + log(1/2)2^(1/2) is -3/2.