I am learning about exponential function from math book for high-school.
I have learned aboutif
and
. I have understand pretty much of it, but I didn't find any explanation if
(why?).
I have tried to understand by myself properties of exponential function.
This is what I think are properties:
1) forfunction is defined but there is no inverse function
2) forfunction is defined but there is inverse function
Can someone explain me a little bit about that?
Hello,Originally Posted by OReilly
I am learning about exponential function from math book for high-school.
I have learned about
I'll try to explain by an example:
Take
a) f(1)=-2 and f(2)=4. That means between x=1 and x=2 the graph of the "function" has to cross the x-axis. But you'll never get the result 0 (zero), if the base isn't zero too.
b)which isn't a real number.
To avoid all these difficulties exponential functions are only defined for positive bases except 1.
Greetings
EB
Yes, I understand now.Hello,
I'll try to explain by an example:
Take
a) f(1)=-2 and f(2)=4. That means between x=1 and x=2 the graph of the "function" has to cross the x-axis. But you'll never get the result 0 (zero), if the base isn't zero too.
b)
To avoid all these difficulties exponential functions are only defined for positive bases except 1.
Greetings
EB
Thanks!
Just out of curiosity, how do you define?
It is generally left undefined. When one defines it though, mostly for the purpose of simplyfing sum- and product notation and power series, it is usually defined as 1. The reason why this is more logical than choosing it to be 0 is because of the real limit of x^x for x going to zero is 1, you may also want to check the graph of the function y = x^x for x tending to 0.Just out of curiosity, how do you define
To define logarithms of negative real numbers you have to go into the complex numbers. The fundamental formula iswhich means that logarithms are really only defined up to the addition of a possible multiple of
. For positive reals, it makes sense to define "the" logarithm as a real number. For negative reals, we can say, for example,
where the logarithm for the positive number x is the usual real value (this definition of log incorporates the famous formula
). But you could equally define it to be
or indeed
if you wanted. You're always going to run up against the problem that now the formula
is only valid up to adding a possible correction of
or a multiple thereof (consider any sensible way of defining
).
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