Question 423397: given log3=x and log5=y express log √(3/5)in terms of x and y
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! log(3) = x
log(5) = y
log(sqrt(3/5) = ?????
let a = 3/5
log(sqrt(3/5) = log(sqrt(a) = log(a^(1/2) = 1/2 * log(a)
substituting for a, we get:
log(sqrt(3/5)) = 1/2 * log(3/5)
since we know that log(3/5) = log(3) - log(5), then this equation becomes:
log(sqrt(3/5)) = 1/2 * (log(3) - log(5))
since we know that log(3) = x and log(5) = y, then this equation becomes:
log(sqrt(3/5)) = 1/2 * (x - y)
if you solve for the original equation and the final equation, you will see that the log of each will be equal to -.110924375
this confirms that the conversion of the formula is correct.
to solve for the original equation, you take the square root of (3/5) and you take the log of it to get -.110924375
to solve for the final equation, you take the log(3) and subtract the log(5) from it and then multiply the result by 1/2 to get -.110924375.
your intermediate answer is that:
log(sqrt(3/5)) = (1/2) * (log(3) - log(5))
your final answer is that:
log(sqrt(3/5)) = (1/2) * (x - y)
this is because x = log(3) and y = log(5) which was given.
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