# How to understand Logarithms, Fundamentally

1. ### delsaber8

47
Alright so my problem is if I don't see exactly how things work (in math) I can't use them. Like what I mean is some people use memorization and other methods but I really need to understand things deep down to be successful.

So my problem is, I have trouble understanding logarithms. Now I haven't gotten to natural logs and stuff, but I have gone over the more basic rules. Unfortunately I have no idea why they work. For example why is logab = loga+logb. Now this kinda goes for all the rules I know, I can use them and apply them but I have no idea what makes them function. If there is anyway to maybe prove them, that would be fantastic.

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2. ### DrewD

490
Do you understand the exponent rules? If you do, extrapolating to logs is a small step. If you have trouble with exponents, start there. Does it make sense to you that ##10^a\cdot10^b=10^{a+b}##?

3. ### delsaber8

47
Yes, in fact just saying that, I'm starting to see the connection.

4. ### DrewD

490
When in doubt, go back to the basic definition of a logarithm. Try to prove a few of the log properties to yourself from the exponent rules. I bet it will be clearer. Sometimes I still go back to check that I haven't forgotten the rules (or math decided to change on me)

5. ### delsaber8

47
Lol, will do. I actually think I have a good grasp now that you showed me the exponential form, I assume everything should fall into place.

### Staff: Mentor

The most basic idea about logarithms is that they are exponents. The log (in some base) of a number is the exponent on that base that produces the number.

For example, the log, base 10, of 100, is the exponent on 10 that yields 100. If you notice that 102 = 100, then 2 is the exponent on 10, the base, that yields 100. In other words, log10(100) = 2.

There is a strong connection between an equation involving a logarithm and another equation that is exponential. If y = loga(x), then x = ay. For a given base, the loga function is the inverse of the exponential function with the same base.

Since these functions are inverses of each other, the composition, in either order, completely undoes the operation of both functions.

loga(ax) = x, for all real numbers x, and
aloga(y) = y, for y > 0

These ideas can be used to verify all of the log properties.

Last edited: Apr 1, 2014
7. ### Jhenrique

686
This is all that you need to know

Exp/log is the bridge between sum and multiplication, the bridge between coefficients/expoents and parcels/factors.

8. ### micromass

19,228
Staff Emeritus
The OP didn't ask for a collection of formulas. The OP asked why the formulas work.

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9. ### Jhenrique

686
The answer "because god wants" is enough for me.

10. ### micromass

19,228
Staff Emeritus
If that's your answer to these kind of questions, then you shouldn't be giving answers to people in the mathematics forum.

11. ### Jhenrique

686
1st, I so sorry.
2nd, I think that no exist a good definition for why the log works, like no exist a intuitive fundamental explanation for sin function or for gamma function. Those set of special funcstions are definied of such that that satisfies arbitrary properties. So, know the properties becomes the focus of attention.

12. ### micromass

19,228
Staff Emeritus
I don't know what you're talking about. There are perfectly acceptable intuitive definitions for the log, the sine and the gamma function. And the properties of these functions can be proven from the definitions perfectly well.

Anyway, let's stop hijacking the thread. This thread is about logarithms and how to prove their fundamental properties. So let's stick to this topic.

### Staff: Mentor

As I said in my earlier post, you can use the definition of a log to prove all of the properties, including the one above.
I'll assume that "log" means "log10" or common log.

Let x = log(a), and let y = log(b).
Then from the definition of the log as an exponent, we have
a = 10x and b = 10y
Then ab = 10x * 10y = 10x + y
Taking the log of both sides, we get log(ab) = log(10x + y) = x + y
But x = log(a) and y = log(b), so log(ab) = log(a) + log(b), which is what was to be shown.

14. ### homeomorphic

Here's one way to think of logarithms. It's a way of scrunching things to include really, really big numbers. This is how the Richter scale for Earthquakes works. Think of log x as being "scrunched x".

Instead of a usual number line that goes

1, 2, 3, 4...

You use one that goes

10, 100, 1000, 10000...

These numbers are meant to be placed at tick marks with equal distances from each other. In the second number line, notice that the numbers are all nicely "scrunched".

In the first number line, you get from one number to the next by adding +1. But in the second number line, you get from one number to the next by multiplying by 10. So, now, multiplication has the same effect as addition. To move to the right by 4 units, you need to multiply by 10^4. The whole point of this scrunched number line is to make this happen. It effectively turns multiplication into addition.

This is the essence of the rule log ab = log a + log b (scrunched ab = scrunched a + scrunched b). That is, if you multiply a by b, you move to the right by log b units on the logarithmic scale.

This used to be very helpful in doing calculations with bigger numbers (before computers). Understanding the slide rule can enhance your logarithmic intuition. Try this wikipedia article:

http://en.wikipedia.org/wiki/Slide_rule

Incidentally, this provides an intuitive reason why x^(1/2) ought to be defined as sqrt(x). When you are using slide rules to find the square root, you find the halfway point.

As multiplication gets demoted to addition, taking natural number powers (repeated addition) gets demoted to multiplication. This is the essence of the rule log a^n = n log a, when n is a natural number, anyway. If you want multiply a by itself n times, on the log scale, you are just going to put n of the scrunched a's (log a) together. Continuing along these lines, you could argue the same thing if we replace n by any rational number (and then with a small leap of faith, perhaps any real number).