Question 492071: Find the exact value for the expression, log5 270 − log5 75 − log5 90
Found 2 solutions by stanbon, Theo:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Find the exact value for the expression,
log5 270 − log5 75 − log5 90
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= log5[270/(75*90)]
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= log5[0.04)
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Cheers,
Stan H.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! your expression is:
log(5,270) - log(5,75) - log(5,90)
in general:
log(a/b) = log(a) - log(b) and:
log(a) - log(b) = log(a/b)
your expression gets transformed using this property as follows:
log(5,270) - log(5,75) - log(5,90)
becomes:
log(5,(270/75/90) which becomes:
log(5,.04)
set y equal to it to get the equation:
y = log(5,.04)
you can convert this to base 10 by using the following formula:
log(5,.04) = log(10,.04) / log(10,5)
now you can use the LOG function of your calculator to solve.
using your calculation, you get y = log(5,.04) becomes y = log(10,.04) / log(10,5) = -2
what this says is that y = log(5,.04) = -2.
the basic properties of logarithms says that:
log(b,x) = y if and only if b^y = x
with your equation, this becomes:
log(5,.04) = -2 if and only if 5^(-2) = .04
5^(-2) is the same as 1/5^2 which is equal to 1/25 which is equal to .04.
this confirms the fact that y = -2 is your answer.
you get:
log(5,270) - log(5,75) - log(5,90) = -2
if you don't do anything except convert log(5,x) to log(10,x) /log(10,5), then you should get the same answer.
your expression becomes:
log(10,270)/log(10,5) - log(10,75)/log(10,5) - log(10,90)/log(10,5) = -2 which becomes:
3.478495142 - 2.682606194 - 2.795888947 = -2.
you get -2 again, confirming that -2 is the correct answer.
you can convert a log from any base to any other base by using the conversion formula.
the general form of the conversion formula is:
log(b,x) = log(c,x) / log(c,b)
this converts a log from the base of b to the base of c.
you make the log of x to the base of b equal to:
the log of x to the base of c divided by:
the log of b to the base of c.
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