What is the mathematical definition for a logarithm?

Question

i.e. for: logb N)

Answer 1

Logaritms are exponents!!!!

Stating that Log b N = x is the same as saying b^x = N.

Answer 2

The natural logarithm is defined in terms of an integral. The Fundamental Theorem of Calculus says that if f(x) is continuous on [a,b], the function F(x)=\int_{a}^x f(t) dt has the property that F'(x)=f(x).

The function y=1/x is continuous everywhere except for 0. So, for x>0, we _define_

ln x = \int_{1}^x 1/t dt

Thus, the domain of ln x is (0,infinity). ln 0 is undefined; ln x is undefined if x is a negative number. (That is the answer to your next question).

Using calculus, one can show that the range of y=ln(x) is (-infinity,infinity) and is one-to-one on its domain so it has an inverse function, which is defined to be y=exp(x). Via _much_ work one can show that exp(x)=e^x, where e is the unique number satisfying ln(e)=1.

I hope this helps! Good luck!

Answer 3

it truly is in certainty a mathematical functionality that enables you to do indices in opposite. case in point, assume we wanted to locate the what capacity you needed to take 3 to to get 9- do you would be able to desire to sq. it, cube it, or what?. Logs may be used to furnish you that huge form. for that reason, the respond develop into tremendously glaring and you need to in all risk have in basic terms guessed it, yet once you had a extra complicated huge form, issues ought to get slightly frustrating in case you probably did no longer have logs.

Answer 4

There is more than one definition, but here is a commonly used one:

ln x = integral from 1 to x of (1/t) dt

Then, log_b x = (ln x) / (ln b)

Answer 5

It's usually defined as the inverse of the exponential function

Answer 6

log_b N =y
is the number to which you need to raise b to get N,
ie.
b^y=N