Question 29231: explain the difference between a logarithm of a porduct and the product of logarithms, give examples of each. Thank you this has me baffled
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! First of all you should learn the correct definition of logarithm of a number.
The logarithm of a real number N is defined only for N strictly positive.
(this means the logarithm of N is not defined for N=0 and not defined for N<0)
Given a positive number N the logarithm of N to a given base b is defined
as the power p to which the base has to be raised to give the number
log N to base b is p implies and is implied by N=(b)^p ----(*)
examples:
1)log(100) to the base 10
Since 100=(10)^2
we write log(100) to base 10 is 2
[here N=100, b=10, so you ask the question 10 raised to what is 100 and you get (10)^2 =100
and so the power is 2 ]
2)log(81) to base 3
Since (3)^4 = 81, we write log(81) to the base 3 is 4
[here N=81, b=3, so you ask the question 3 raised to what is 81 and you get (3)^4 =81
and so the power is 4 ]
Got it!
3)If you are given (2)^5=32, how do you express it in the logarithmic language?
Notice that our N=32, b=2 and p=5 [look at (*)!(asking you to see stars!)]
Therefore log(32) to the base 2 is 5
4)If you are given (5)^3= 125, how do you express it in the logarithmic language?
Notice that our N=125, b=5 and p=3 [look at (*)]
Therefore log(125) to the base 5 is 3
If it is confusing, keep chanting the definition (*)
Cool is n't it?
Four important initial formulae
caution:Every formula in logarithm for any consideration is applicable only for a particular given base.[If you are changing base then you have to apply the formula for the change of base]
i)for a given base b
log(m) +log(n) = log(mn)----(1)
log(m) -log(n) = log(m/n)----(2)
n[log(m)] = log[(m)^n]----(3)
log(m) to a base (b) given by {[log(m) to a base (c)]/[log(b) to the base (c)]
where c is the new base.
Note: logarithm of any number (m) to the same base (m) is 1
as(m)^1=m
Note: NO FORMULA for log(m)Xlog(n).
That is for a given base b there is no formula for multiplication between log(m) and log(n)
Note: log(m) to the base (a) multiplied by log(a) to the base (n)
that loga(m) X logn(a) = logn(m) -----(*)
and this should not be mistaken for a stadard formula for multiplication between two logartihms. What you must NOTE here is though two logarithms are multiplied they are not with respect to the same base and all formulae in logarithms are dealing in
logarithm of a product = sum of logarithms, (for a given base b)
logarithm of a quotient = difference between the logarithm
(for a given base b) and the other two formulae too are for a given base b
and in the star (*) above one logarithm has for its base a and the other has for its base n
This (*) if you look closely,is the result of change of base formula
Actually this is the result of logn(m)= loga(m)/loga(n)
(Change of base rule) (and here the new base is a)
Multiplying by loga(n)
we get (*)
Examples of formula (1)
When it is mentioned "common logarithms" the base is 10
You use this for numerical problems
When it is mentioned "natural logarithms" the base is e (the irrational number more than 2 and less than 3 called the Napierian base named after the mathematician Napier)
You use this for problems in Calculus.
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