Please help. I need to solve this equation:

e^x = 5x - 3

Thank you.

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- May 24th 2006, 07:56 PMPacManFroggComplicated Logarithms
Please help. I need to solve this equation:

e^x = 5x - 3

Thank you. - May 24th 2006, 09:15 PMearbothQuote:

Originally Posted by**PacManFrogg**

by your equation you calculate the intercepting points of graphs of an exponential function and a line. I've attached a diagram to show the situation.

I don't know a method to solve this equation algebraically. Instead I would use an iteration(?) (I mean that you have to construct a convergent sequence to obtain the solution). I'll demonstrate a method which I know under the name Newton's method:

1. Transform the equation into a function: f(x)=e^x-5x+3

2. Choose a x-value for a start. Here I would say: $\displaystyle x_0 = 1$

3. The next x-value is calculated by:

$\displaystyle x_{n+1}=x_n-{f(x_n)\over f'(x_n)}$ where f'(x) is the first derivation of f. With your problem it is: f'(x) = e^x-5

So you get:

$\displaystyle x_{1}=1-{{e^1-5+3}\over {e^1-5}}\approx 1.3147...$

Now plug in this value for $\displaystyle x_n$ and you'll get the next x-value which comes closer to the solution:

x_1 = 1.3147...

x_2 = 1.4323...

x_3 = 1.4653...

x_4 = 1.4687...

x_5 = 1.4688...

x_6 = is the same as x_5. That means you substract zero from the previous x-value which is only possible if f(x) = 0. That was intended. So you've found one solution.

Change the x_0-value to 2 and you'll get the second solution: x = 1.74375...

Greetings

EB - May 24th 2006, 09:37 PMCaptainBlackQuote:

Originally Posted by**earboth**

RonL - May 24th 2006, 09:54 PMearbothQuote:

Originally Posted by**CaptainBlack**

you are right.

But as I've mentioned above it is x = 1.74375...

Greetings

EB - May 24th 2006, 10:21 PMCaptainBlackQuote:

Originally Posted by**earboth**

The problem can also be solved using the Lambert W function.

RonL - May 24th 2006, 10:56 PMearbothQuote:

Originally Posted by**CaptainBlack**

Hello,

and äääähem. Well I've heard the name of Heinrich Lambert. But I don't know his famous W-function (I believe to remeber that it deals somehow with complex arguments and differential equations(?)), and worse: I wouldn't be able to explain how to use.

Greetings

EB - May 25th 2006, 04:03 PMThePerfectHackerQuote:

Originally Posted by**earboth**

- May 25th 2006, 10:06 PMCaptainBlackQuote:

Originally Posted by**ThePerfectHacker**

x involved is not even >-1/e) so we are in a region where W is multi-valued

(as we should expect since there are two real roots to this problem).

My mention of W was a sort of joke, theoretically it can be used to write

the solution in closed form, practically one is better off using numerical

methods to solve the problem.

RonL