Understanding Logarithms?

Question

What exactly is a Logarithm? and what is it used for? An example of how to use Logarithms would help alot.

By understanding Logarithms I mean like... for example, Y=mx+b is describing a line. M is the slope and B is the Y-intercept. I'm trying to understand Logarithms in the same way.

Answer 1

First, logarithms are not birth control methods used by lumberjacks (um, old math joke... sorry about that...)

IMO, logarithms are most useful when you are dealing with large numbers. For example, if you need to know about how big 2^100 is, you could spend all day calculating it, or you could do a logarithmic calculation that takes a few minutes: log 2 = .30103, therefore log 2^100 = 30.103, so 2^100 is a somewhat more than 10^30.

The point to logarithms is that you are dealing with exponents, i.e. log 2 = .30103 means 10^.30103 = 2. So, to multiple two numbers, you add their exponents, while to find the power of a number, you multply the logarithm by the power. (This may sound like Greek right now, but it will make sense when you understand logarithms better.)

I guess it should also be mentioned (as ocdscale does) that certain types of measurements are done on a logarithmic scale because they make more sense. For example, the Richter scale used to measure earthquakes is a logarithmic scale.

Answer 2

You know what exponents are: 5^3 = 5*5*5. 3^4 = 3*3*3*3, etc. Logarithmns are just exponents worked in reverse. Use this definition:

Log[base b](a) = c is the same as saying a = b^c

The base of a log is usually written as a subscript, but "base b" was as best I could write it here. If you don't see a base number, then it's assumed the base is 10. For example, Log(1000) = 3. Here are some other examples:

Log[base 4](16) = 2
Log[base 5](25) = 2
Log[base 2](8) = 3
Log[base 10](19) = 1.2787536... (not an exact exponent)
Log[base 3](-27) = no answer, because nothing raised to the 3rd power can be a negative number.

If you were to graph y = log(x), it would look like this:
http://mathworld.wolfram.com/CommonLogarithm.html

Notice that you can't take the log of a negative number, because anything raised to a power must be a positive number. And smaller and smaller positive numbers must be results of the base raised to some negative exponent (1/10 = 10^-1, 1/100 = 10^-2, etc.) so you get an asymtope at x=0.

Answer 3

Logarithms are tricky things. They are really useful though when you get into Chemistry and if you're really nerdy, radio communications.

Log is naturally assumed is base 10. Base ten meaning 10 to the what power equals this. This may help:
log 100 = 2 -- 10 to the what power equals 100? 2, of course.

I know it's complicated, but check this place out: http://mathworld.wolfram.com/Logarithm.html

Answer 4

It's difficult to describe logs in here because we can't use subscripts (The small numbers on the bottom), but here goes:

Log[small 2] 32 = 5
What does this mean?
It means if you are 'thinking' in terms of 2s (the small number, it's called the Base), to get to 32 you need to multiply 2 by itself 5 times.

Log[small 2] 32 = 5 is the same as saying 2^5 = 32

It's difficult for me to think of a real life use for logs though.
The only practical use off the top of my head is as a scale for certain measurements. On a logarithmic scale, the distance from 1 to 10 is the same as 10 to 100 which is the same distance as from one million to ten million. It can be useful to graph (really) small things alongside (really) large things.

Answer 5

if Log (base b) A = X , then b^x = A

log (base 2) 8 = 3 because 2^3 = 8

logs are useful in solving problems where the unknown is an exponent. For example: 3.7 ^ x = 42.5
1) take log of both sides : X log 3.7 = log 42.5
2) X = log 42.5 / log 3.7
3) X= 2.865
4) check: 3.7 ^ 2.865 = 42.5

Heres a practical example: How long will it take your money to triple earning 7.5% ?
the answer: 3 = (1.075)^t where t is the time to triple.
1) log 3 = t log 1.075
2) log 3 / log 1.075 = t
3) t = 15.2 years