Subject: natural logarithm functions
The following two questions are some of my son's homework that he is having trouble with......any advice or assistance would be appreciated.
(eX)5=1000.............the X and 5 are exponents
lnx + ln(x+3) = ln10
There are some basic facts that you have to know about logs and exponentials to solve these problems. First the natural logarithm and exponential functions are inverses, that is
Also there are three properties of logarithms that are very useful,
ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b) and
For your first problem that take the natural logarithm of both sides to get
, and hence 5x = ln(1000). Using the properties of logarithms this can be written in a simpler form as so x=(3/5) ln(10).
For your son's second problem use the properties of logarithms with
ln(x) + ln(x + 3) = (10) to get ln(x(x + 3))= ln(10).
Now take the exponential of both sides and use the fact that the exponential and natural logarithm functions are inverses to arrive at x(x + 3) = 10. This expression can now be expanded to yield a quadratic equation in x that has two solutions.
You need now to take these values of x and verify that they are solutions to the original problem. The reason that this is important here is that ln(x) is only defined for positive x.
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