How do logarithms work?

Question

where does the LN function come from and what does it do.

Answer 1

Well it reverses power functions.

So for instance, if 10^6=1000000, then log 1000000 = 6. The base number is always 10 unless stated otherwise.

Here is an example of a log of different base:

If 2^4=16, then log (base 2) 16 = 4.

So in general, if n^x=y, then Log (base n) y = x.

When you get into calculus, you will learn about a special number called e.
e = 2.71828182845905
Anytime e^x=y, and you want to solve for x, instead of saying "log (base e)" you say "Natural Log" which is abreviated LN.

So if e^x=y, then LN y = x.

Answer 2

Logarithms are the inverses of the exponentials. It is devised to solve for certain variables, which are on exponents.
E.g.
in 2^x = 32, one could easily show that x = 5, because
2^x = 32
2^x = 2^5
x = 5
but how are you going to solve for x in the exponential equation
2^x = 3 ?
Thus they created a notation for an exponent.

Logarithm:
if
a^b = c, then
b = log_a c
The general solution for the exponent. in our example 2^x = 3, the solution is x = log_2 3 ≈ 1.58.

In
a^b = c, a is the base, b is the exponent and c is the power. So,
b = log_a c,
a is still the base, b is still the exponent and c is still the power. Therefore, the whole logarithm is just an exponent and
a^(log_a c) = c
Actually, log_a c is read " the logarithm of c to base a".

The log function is available for any base > 0 and ≠ 1.

The ln function (natural logarithm) is still a logarithm. It has a unique base, the irrational number e. Therefore,
ln x = log_e x

The ln function, though it has a seemingly useless base, is very useful esp. in higher maths like calculus, just like the "natural exponential" e^x.

^_^

Answer 3

How Do Logarithms Work

Answer 4

Logarithms were developed long before 'e' was discovered.

ln is called the natural logarithm.

Logarithms do a lot of things like appear in mathematical equations describing the world, and making certain tasks easier; which was a reason they were thoroughly investigated at one point in time

See wiki for more info...

Answer 5

The natural logarithm function is the inverse function of (e^x). Where e is Euler's number (non-terminating, non-repeating constant ... like pi ... approximately 2.71828).

If we define f(x) = e^x, then

f(1) = e


If we define g(x) = ln x, then

g(e) = 1


... as they are inverses of each other, f( g(x) ) = x and g( f(x) ) = x.


e^x is important, as its derivative is itself :

f(x) = e^x, then f'(x) = e^x, f''(x) = e^x ... etc.

Answer 6

logarithms were initially developed to simplify multiplication of large numbers.

this is back in the day, 1600s to 1700s, when they we trying to calculate the orbit of planets and such.

Answer 7

I solves equations like

(1.035)^x = 2
x = ln(2) / ln(1.035)

Magic.

Answer 8

i have been asking that question for years. If I knew that, I would not have flunked out of calculus.