log8 4(square root sign) 1/2
log8 4√(1/2)
recall that logaa^k = k
so we have to change 4√(1/2) to base 8
4√(1/2)
= 4/√2
= 4/√2 (√2/√2)
= 2√2
= (8^(1/3)) (8^(1/6))
= 8^(1/2)
so log8 4√(1/2)
= log8 8^(1/2)
= 1/2
To simplify the expression log8 4 √(1/2), we can start by simplifying the contents of the logarithm.
1. Simplify the square root of 1/2:
√(1/2) = √(1) / √(2) = 1 / √(2)
2. Express the denominator (√(2)) in exponential form:
1 / √(2) = 1 / (2^(1/2))
3. Convert the logarithmic expression using the logarithmic identities:
log8 4 (1 / (2^(1/2))) = - log8 4 (2^(1/2))
4. Use the logarithmic property that states loga (b^c) = c * loga (b):
- log8 4 (2^(1/2)) = - (1/2) * log8 4 (2)
5. Express 4 and 8 as powers of 2:
- (1/2) * log8 4 (2) = - (1/2) * log2 2 (2) / log2 2 (8)
6. Simplify the logarithm:
- (1/2) * log2 2 (2) / log2 2 (8) = - (1/2) * 1 / 3 = - 1/6
Therefore, log8 4 √(1/2) simplifies to -1/6.
To simplify the expression log8 4(sqrt(1/2)), we can apply logarithmic properties and simplifications. Here's how to do it step by step:
Step 1: Simplify the square root (sqrt) of 1/2.
sqrt(1/2) = sqrt(1) / sqrt(2) = 1 / sqrt(2)
Step 2: Rewrite log8 4(sqrt(1/2)) as an equivalent exponential expression.
Let's call the value we are looking for x:
8^x = 4 * (1/sqrt(2))
Step 3: Simplify the right side of the equation.
Remember that dividing by a square root is equivalent to multiplying by its reciprocal:
8^x = 4 * (1/sqrt(2)) = 4/sqrt(2) = 4 * (1/2^(1/2)) = 4 * (1/√2)
Step 4: Rationalize the denominator of the right side.
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator:
8^x = 4 * (1/√2) * (√2/√2) = 4 * √2 / 2
Step 5: Simplify the right side of the equation.
8^x = (4 * √2) / 2 = (2 * 2 * √2) / 2 = 2 * √2
Step 6: Rewrite the equation using exponential notation.
8^x = 2 * √2 can be written as log8 (2 * √2) = x
So, the simplified expression is log8 (2 * √2) = x.