Why are logarithmic functions so complicated? Like understanding them without using numbers?

Question

Any good sites or advice?

Answer 1

Logarithms are very tricky, especially when you first start out with them. The way I am best at interpreting them is by realizing their relationship to undo exponentiation. You probably all ready know the expression

log_a x = y <==> a^y = x

This means that the two equations above are equivalent. For instance, if a=10, x = 100, then y = 2. So when given a problem like y=log_2 8, you should think to yourself, "What exponent of two will bring it to 8?"

Hopefully you will eventually conclude that y=3 since 2 cubed is eight.

That all being said, logarithms are tricky business. However, they are incredibly important, and if you go on to study math more rigorously, when you study the lovely number "e" and it's superb functions, e^x and ln x (or log_e x, if you prefer), you will see that those functions pop up in nature and science in the most unpredictable places.

Good luck with the logs!

Answer 2

They may seem mysterious at first, but if you remember some basic rules, you will have no trouble.

The logarithm (L) to the base (a) of a number N = L if a^L=N.
So log_a N = Lif a^L= N. The two most common bases are 10 and the number e.

If the base is 10 we just write log N = L (base = 10 is understood). Likewise if the base is e we write ln N= L (base e is understood).

Here are some very important laws of logarithms you should commit to memory:
log_a (1) = 0
log_a (a) = 1
log-a(M*N) = log_a (M) + log_a (N)
log_a(M/N) = log_a (M) - log_a (N)
log_a M^1/n = (1/n) log_a(M)
a^(log_a (x)) = x
log_a (a^x) = x
These laws hold for any base but are usually for bases 10 or e.
Try these laws out and you'll soon become proficient with logarithms and exponential equations. Remember the log function is not defined for N <= 0.

Answer 3

Logarithmic functions are not that complicated. They are defined as the inverse function for exponential equations like

a^b = c given a is a constant. So if you had a curve

y(x) = a^x, its just a curve like y(x)=x^2 (not the same curve, but a curve all the same, right?)

then the log function is just its inverse. Traditionally, you could find the inverse of y=a^x by swapping x's and y's

x = a^y
but now you want to rewrite this as y(x) again (this new y(x) is the inverse of the y(x) given above) but how do you get the y out of the exponent? Thats where logarithms come in.
One of the properties of logarithms is

log(a^n) = log (a*a*a...*a) [n a's] = log(a) + log(a) + ...+ log(a) [now n log(a)'s] ] n*log(a)

so applying this above,

log(x) = log(a^y) = y*log(a) so y(x) = log(x)/log(a)

if you define your log in base a i.e. log_a then log(a)=1 and you get y(x) = log(x). So logarithms are just another inverse function, like y(x) = sqrt(x) is the inverse of y(x) = x^2.