Question 492071
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.