# Ln and exp

1. May 10, 2013

### eng.mustafaS

hi every one ..
i have some question
1 - why the value of exp = 2.71 ?? were from they get this value ? any derive or rule ?

2 - who i can fiend the value of ln and exp and sin or cos with out using calculator ?

like cos 56
sin 28 ??

or

ln 24 ?
exp 33 ?
without using calculator ?

thank you

2. May 10, 2013

### Staff: Mentor

e is a most curious number as a base.

e^x is the only curve whose derivative with respect to x is itself. This fact is used a lot in solving differential equations by assuming solutions based on e^x for certain types of differential equations.

You can read more about it at:

http://en.wikipedia.org/wiki/E_(mathematical_constant)

Its true that its not an easy number to work with numerically but the insight gained in using it is far more profound.

Last edited by a moderator: May 10, 2013
3. May 10, 2013

### Staff: Mentor

You can calculate e by yourself, using this sum:

$$e = \sum\limits_{n=0}^\infty \frac 1 {n!}$$

Edit: and a spreadsheet.

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4. May 10, 2013

### Mentallic

You can find some relatively reasonable approximations to these values using your basic rules, but if you need any sort of precision or are short on time, you'd definitely want to use a calculator.

For the logs, you can split them up into a sum of smaller logs:

$$\ln(24)=\ln(2^33)=\ln(2^3)+\ln(3)=3\ln(2)+\ln(3)$$

Now, you just need to have ln(2) and ln(3) memorized to make the calculation.

For exponentials:

$$e^{33}=10^{\log_{10}(3)\cdot 33}$$

Where

$$\log_{10}(3)=\frac{\ln(3)}{\ln(10)}=\frac{\ln(3)}{\ln(5)+\ln(2)}$$

So, if you memorize the natural log of 2,3,5 and 7, you'll have most values covered, so then you can find an approximation to $\log_{10}(3)$ and thus will have the order of magnitude of e33.

The trigs are trickier and I wouldn't recommend making a calculated approximation for them. As long as you understand the graphs, you should be able to make reasonable approximations in your head.
If you still want further precision though, in degrees,

$$\cos(56)=\cos(45+11)$$

$$=\cos(45)\cos(11)-\sin(45)\sin(11)$$

Now, $\cos(45)=\sin(45)=1/\sqrt{2}$
and for small angles x in radians, $\cos(x)\approx 1$ and $\sin(x)\approx x$.

Thus we will take $\cos(11)\approx 1$
and (remembering to convert 11o to radians), $\sin(11^o)=\sin(\frac{11\pi}{180})\approx \frac{11\pi}{180}\approx \frac{35}{180}\approx 1/5$

So finally,

$$\cos(56)\approx \frac{1}{\sqrt{2}}(1-1/5)=\frac{4\sqrt{2}}{10}\approx \frac{2\cdot \frac{7}{5}}{5}=14/25= 56/100 = 0.56$$

Where the true value is $\cos(56)=0.559...$

As you can see, it's a lot of work for this approximation.

Also, I'd like to note that you shouldn't expect your approximations to come this close in future (0.56 compared to 0.559... is less than 0.1% off), it's just that my over-approximation of cos(11o) and taking away my over-approximation of $\frac{11\pi}{180}\approx 1/5$ lead to a very close result by pure luck.

Last edited: May 10, 2013
5. May 10, 2013

### HallsofIvy

Staff Emeritus
Another way of looking at it:

For any positive number, a, if $y= a^x$ then dy/dx is given by
$$\lim_{h\to 0}\frac{a^{x+h}- a^x}{h}= \lim_{h\to 0}\frac{a^xa^h- a^x}{h}= a^x\lim_{h\to 0}\frac{a^h- 1}{h}$$

That is, the derivative of $a^x$ is just $a^x$ itself times a number- that number being $\lim_{h\to 0}(a^h- 1)/h$.

It is easy to see that if a= 2, $\lim_{h\to 0}(2^h- 1)/h$ is less than 1. For example, with h= 0.0001, that fraction is $(2^{0.0001}- 1)/.0001= (1.000069317- 1)/.0001$= 0.000069317/.0001= .69317

And if a= 3, $\lim_{h\to 0}(3^h- 1)/h$ is larger than 1. With h= 0.0001, again, we have $(3^{0.0001}- 1)/.0001= (1.00010987- 1)/.0001$= .00010987/.0001= 1.0987.

If you do the same thing with, say, a= 2.5 and a= 2.8, you would again get that constant less than 1 for 2.5, larger than 1 for 2.8 but both closer to 1. There exist a value of a in between 2 and 3, and between 2.5 and 2.8, such that the constant is exactly equal to 1: that is, for that a, the derivative of $a^x$ is just $a^x$ itself. We define "e" to be that number.

Last edited: May 10, 2013
6. May 10, 2013

### Integral

Staff Emeritus
Before calculators we used tables for trig functions and logs. Fact is there is no easy way to compute these function values without some form of an aid.

7. May 10, 2013

### eng.mustafaS

what is "n" ?

pleas can you fiend for me the value of e^4 by this equation
$$e = \sum\limits_{n=0}^\infty \frac 1 {n!}$$

8. May 10, 2013

### phinds

The sum process (with n being the element index) does not compute powers of 3, it computes e. You'll have to multiply it by itself 4 times to get e^4

What grade are you in? I get the impression that you are asking about things you understand so little that you can't even understand the answers. That just means you have to study the basics a bit more to get rid of your confusion.

9. May 10, 2013

### eng.mustafaS

thank you very much

10. May 10, 2013

### eng.mustafaS

i am in college .. but i am not good in maths

and tell now i dont know how i fiend e^4

11. May 10, 2013

### phinds

e^4 is e*e*e*e

12. May 10, 2013

### micromass

Staff Emeritus
$$e^4 = \sum_{n=0}^{+\infty} \frac{4^n}{n!}$$

13. May 10, 2013

### Staff: Mentor

n takes all possible integer values, starting with 0.

Not by this equation, but you can use this one:

$$e^x = \sum\limits_{n=0}^\infty \frac {x^n} {n!}$$

The first equation was just a special case, for x=1 (and obviously e1 = e).

14. May 10, 2013

### skiller

I'm confused here. Why have both of you used a "3", when surely you mean "e"? If it was just one of you then I would simply put it down to a typo, but as two different people have done this, then maybe I've completely misunderstood something. Apologies if that is the case!

15. May 10, 2013

### HallsofIvy

Staff Emeritus
To err is human- to really screw up requires a computer! I suspect Mentalic made a typo then phinds "copied and pasted".

16. May 10, 2013

### skiller

Haha! I've known that proverb all my life and it's still one of my favourites.

I agree that would be a reasonable explanation normally, but it looked to me that Mentallic's wasn't just a typo as he/she continued to to say log(3) etc later on. And phinds's message certainly doesn't appear to have been of any copy'n'paste effect to me.

Like I say, it really made me question myself. But glad we've agreed that it should've been an "e" in both cases - it certainly has satisfied me that I'm not going mad.

I wasn't trying to be a pedant, but on sites such as this, little things like these can confuse a person who is trying to learn - not just the original poster. Sorry if I caused any offence by pointing it out.

17. May 10, 2013

### phinds

Uh ... stupidity ? Wait, no that's not it. It's a typo I tell you, a typo. Or a senior moment. I forget.

18. May 10, 2013

### skiller

Haha! My vote is for "senior". I consider myself senior too, but by no means consider myself "senior" to any of your maths knowledge. I'm well over half my way to my late 80s.

19. May 10, 2013

### Mentallic

Ahaha yeah that was a terrible [STRIKE]typo[/STRIKE] senior moment

Of course, my post should have said

$$e^{33}=10^{\log_{10}(e)\cdot 33}$$

And everything else should be changed accordingly. I'd edit it, but that option is gone now.

Sorry!

20. May 10, 2013

### skiller

Good job I'm here then!