Question 428054: How do you expand Logarithm, log base5 (m exponent 4 * n exponent 3)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! log(5,m^4*n^3) is what i believe you are asking about.
note that log(b,x) means the log of x to the base b.
in your case, the formula i showed above means log of (m^4*n^3) to the base 5.
i think you would solve this as follows:
log(5,m^4*n^3) = log(5,(m^4)*(n^3))
since log (a*b) = log(a) + log(b), then this should be equivalent to:
log(5,m^4) + log(5,n^3)
since log (a^b) = b * log(a), then this should be equivalent to:
4 * log(5,m) + 3 * log(5,n)
in order to confirm that we did this correctly, we need to solve this for specific values of m and n, using both the original expression and the final expression, to see if the generality holds.
we'll take random values of m and n and see where that leads us.
let m = 5 and n = 7
our original expression is:
log(5,m^4*n^3)
when m = 5 and n = 7, our expression becomes:
log(5,5^4 * 7^3) which becomes log(5,625 * 343) which becomes log(5,214375).
log(5,214375) = log(10,214375) / log(10,5) = 7.627185865
now we'll move to our final expression of:
4 * log(5,m) + 3 * log(5,n)
when m = 5 and n = 7, this expression becomes:
4 * log(5,5) + 3 * log(5,7).
log(5,5) = log(10,5) / log(10,5) = 1
log(5,7) = log(10,7) / log(10,5) = 1.209061955
our expression of 4 * log(5,5) + 3 * log(5,7) becomes 4 * 1 + 3 * 1.209061955 which becomes 4 + 3.627185865 which becomes 7.627185865.
our original expression yields an answer of 7.627185865.
our final expression yields an answer of 7.627185865.
the answer are the same leading to the conclusion that we converted the original expression to the final expression correctly.
the log conversion formula that we used to help confirm the answer is correct is:
log(a,x) = log(b,x) / log(b,a)
this means that the log of x to the base a is equal to the log of x to the base b divided by the log of a to the base b.
an example is log(5,625).
we know that log of 625 to the base of 5 is equal to 4 because 5^4 = 625.
this is because of the basic logarithmic rule that says:
y = log(a,x) if and only if x = a^y.
to use our calculator to confirm that we use the conversion formula of:
log(5,625) = log(10,625) / log(10,5).
the numerator is the log of the same number only to the base of 10 rather than 5.
the denominator is the log of the base of 5 taken to the base of 10.
now we can use our calculator to solve.
the calculator gives us log(5,625) = 2.795880017 / .698970004 = 4
we got the answer for log to the base of 5 by using our calculator which gives us logs to the base of 10 and the conversion formula that allows us to do so.
your answer, if i understood the question correctly is:
log(5,m^4*n^3) is equivalent to 4 * log(5,m) + 3 * log(5,n)
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