SOLUTION: Given: log5=a, log3=b,log2=c. Find: log

SOLUTION: Given: log5=a, log3=b,log2=c. Find: log_30(8)

Question 1148108: Given: log5=a, log3=b,log2=c. Find: log_30(8)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you are given:

log(5) = a
log(3) = b
log(2) = c

you want to find log_30(8).
you can use the log base conversion formula to get:
log_30(8) = log(8) / log(30)

we start with log_30(8) = log(8) / log(30).
8 is equal to 2^3 and 30 is equal to 5 * 3 * 2, so the formula becomes:
log_30(8) = log(2^3) / log(5 * 3 * 2)
by the rules of logarithms, this becomes:
log_30(8) = 3 * log(2) / (log(5) + log(3) + log(2))
since log(5) = a and log(3) = b and log(2) = c, we get:
log_30(8) = (3 * c) / (a + b + c)

your answer should be:

log_30(8) = (3 * c) / (a + b + c)

to see if that's true, we replace a, b, c with their respective values of:
a = log(5)
b = log(3)
c = log(2)
to get:
log_30(8) = 3 * log(2) / (log(5) + log(3) + log(2)) = .6113851413.

you can use your calculator to see that log(8) / log(30) = the same value.

you can also use your calculator to see that log_30(8) = .6113851413 if and only if 30 ^ .6113851413 = 8.
it did equal that on my calculator.

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