Use properties of logarithms to rewrite the expression. Simplify the results if possible.
Assume all variables present positive real numbers.
log2(2squareroot 3/5)
log2(2√3/5))
= log2(2) + 1/2 log2(3/5)
= 1 + 1/2 (log2(3)-log2(5))
To rewrite the expression using properties of logarithms, we can utilize the following three properties:
1. The logarithm of a product is equal to the sum of the logarithms of the individual factors: log(a * b) = log(a) + log(b).
2. The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator: log(a / b) = log(a) - log(b).
3. The logarithm of a number raised to a power is equal to the product of the exponent and the logarithm of the number: log(a^b) = b * log(a).
Now let's apply these properties to the given expression:
log2(2 * √3/5)
First, let's simplify the square root:
√3/5 = √(3/5)
Next, we can use the property that the square root of a number is equal to the number raised to the power of 1/2:
√(3/5) = (3/5)^(1/2).
Applying the third property, we have:
log2(2 * (3/5)^(1/2))
Now, we can simplify further using property 1:
log2(2) + log2((3/5)^(1/2))
Since log2(2) is equal to 1, we can simplify this expression to:
1 + log2((3/5)^(1/2))
Finally, applying the third property again, we have:
1 + (1/2) * log2(3/5)
So, the expression log2(2√3/5) can be rewritten as 1 + (1/2) * log2(3/5).