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Hey, the question is:
7(i) Express each of the following in terms of log_10 x and log_10 y.
(a) log_10 (x/y)
(b) log_10 (10x^2y)
(ii) Given that
2 log_10 (x/y)= 1+ log_10 (10x^2y)
find the value of y correct to 3 decimal places.
If someone could help me with this i'd be most grateful! :D
Laws of logs:Quote:
Originally Posted by LoveDeathCab
log_c(a) = b means that c^b=a
log(a.b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^b) = b log(a).
So:
(a) log_10 (x/y) = log_10(x) - log_10(y)
(b) log_10 (10x^2y) = log_10(10) + log_10(x^2 y) = log_10(10) + log_10(x^2) + log_10(y)
........................... = log_10(10) + 2 log_10(x) + log_10(y)
Now from the definition of logs log_a(a) = 1, so log_10(10)=1, so:
log_10 (10x^2y) = 1 + 2 log_10(x) + log_10(y)
(ii) Given that
2 log_10 (x/y)= 1+ log_10 (10x^2y)
using part (a) above on the left hand side, and (b) on the right hand side:
2 [log_10(x) - log_10(y)] = 1+1 + 2 log_10(x) + log_10(y) = 2 + 2 log_10(x) + log_10(y),
so:
-3 log_10(y) = 2
or: log_10(y) = -2/3, so y=10^(-2/3) or : y ~=0.215.
RonL