log (sqrt4(x^16 / y^4))
What is the value of the expression when logx=8 and logy=1?
My work:
x=10^8 y=10
I tried to plug these in but my calculator but it said "overflow." Is there an easier way to do this?
I assume sqrt4(n) means 4th root. So,
log 4throot(x^16/y^4) = log(x^4/y)
= 4logx - logy
= 4*8 - 1
= 31
Of course, you could have worked it without extracting the 4th root:
log 4throot(x^16/y^4)
= 1/4 log(x^16/y^4)
= 1/4 (16logx - 4logy)
= 1/4 (16*8-4*1)
= 1/4 (128-4)
= 1/4 * 124
= 31
To evaluate the expression log(sqrt(4(x^16 / y^4))), we can simplify the expression step by step.
Step 1: Simplify the expression inside the square root.
sqrt(4(x^16 / y^4)) becomes sqrt(4 * (x^16) / (y^4)).
This simplifies to 2 * (x^8 / y^2).
Step 2: Simplify the expression inside the logarithm.
log(sqrt(4(x^16 / y^4))) becomes log(2 * (x^8 / y^2)).
Using logarithmic properties, we can rewrite this as log(2) + log(x^8) - log(y^2).
Simplifying further gives log(2) + 8 * log(x) - 2 * log(y).
Step 3: Substitute the given values.
Given that log(x) = 8 and log(y) = 1, we can substitute these values into the expression.
Thus, log(2) + 8 * log(x) - 2 * log(y) becomes log(2) + 8 * 8 - 2 * 1.
Step 4: Calculate the expression.
log(2) is a constant whose approximate value is 0.30103.
Therefore, the expression becomes 0.30103 + 8 * 8 - 2 * 1.
Calculating the values gives us 0.30103 + 64 - 2, which simplifies to 62.30103.
Hence, the value of the expression log(sqrt(4(x^16 / y^4))) when log(x) = 8 and log(y) = 1 is approximately 62.30103.