Question

Anyone knows how to solve this??

Answer 1

\[\frac{ (x + 5) }{ 3 } = \frac{ \ln(x) }{ -2 }\]

Answer 2

I'm really confused how to go on about this. I tried doing it like this:

\[-2(x + 5) = 3\ln(x)\]
\[-2x -10 = \ln(x^3)\]
\[e ^{-2x -10} = x^3\]

and after I'm just stuck.. ._.

Answer 3

i forget how to do this sry

Answer 4

hint:
Examine what you just wrote:
−2x−10=ln(x3)

Answer 5

actually : \(−2x−10=ln(x^3)\)

Answer 6

hint:
graph (by hand) each side of the equation
\(−2x−10=ln(x^3)\)

Answer 7

well it does imply that e has to have an exponent of -2x-10 to give x^3

so \[e ^{-2x}*\frac{ 1 }{ e ^{10} } = \ln(x^3)\]

Answer 8

Do they ever intersect?

Answer 9

\[\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\]Choose a suitable approximation.

Answer 10

@mathmate

yes they do intersect which means that they have one 1 root but I want to find the value for x in order to find the coordinate of the intersection.

Answer 11

@Audio I'm not sure what that is, can you please expand on that?

Answer 12

It is, indeed, an expansion. :)

http://mathworld.wolfram.com/SeriesExpansion.html

Answer 13

The best bet here is to graph though. You never know how inaccurate an answer can be when you use Taylor Expansions to solve transcendental equations.

Answer 14

@nopen Are you working on calculus or numerical methods?

Answer 15

um the working that you see in my first comment is what I used to approach this problem.

Answer 16

This is basically a numerical methods problem, in which you know there is a root, and need a method to refine the value accurately. This is the approach.

Answer 17

ah, yes that's exactly what I wanted to do actually. But I don't know really know how, as I'm stuck at the 3rd step

Answer 18

The third step is the meat of the problem.
start with something simple,like
f(x)=-2(x+5)-3log(x)
and calculate f(0.01) and f(0.05) to confirm that there is a root.

Answer 19

I'm getting
f(0.01) = -4.02
and for f(0.05) = -6.196

Answer 20

but wait how are you taking random numbers for the domain like that?

Answer 21

Sorry, I usually use log(x) to mean ln(x), and log10(x) when it's to base 10.
We use log10 so rarely that I use log to mean log e.

Answer 22

They are not random numbers, they are obtained from the graph that you said earlier that there is a root.

Answer 23

From graphing, we know the value of x is very small, such that -2(x+5) is about -10.

Answer 24

oh. Oops okay

then f(0.01) = 3.79
and f(0.05) = -1.11

Answer 25

Anyway, have you recalculated f(0.01) and f(0.05), which should have alternate signs.

Answer 26

Good!

Answer 27

yup

Answer 28

Now, we can do one of the two things.
1. by trial and error, or bisection method:
Keep calculating f(x) while narrowing down between 0.01 and 0.05 until you get f(x) as close to 0 as you wish. You'd better have a programmable calculator or a computer for that.

Answer 29

For example,
f(0.01)=3.795510557964274
f(0.05)=-1.112803179338027
f(0.035)=-0.01277834752183082
make guesses along the way
f(0.03465)=0.01807266003867447
f(0.03475)=...

Answer 30

are you still with me?

Answer 31

Yes, but I see that's gonna be a long process by hand :S

Answer 32

Good, now you will appreciate
2. by Newton's method.

Answer 33

2. Newton's method says
given
f(x)=-2(x+5)-3log(x)
and an initial approximation x0,
under certain conditions of convergence (which we satisfy here),
a better approximation
x1=x0-f(x0)/f'(x0)
where f'(x)=d f(x)/d x = -2 - 3/x in this case

Answer 34

and we can repeat the process until \(x_n\) is identical to \(x_{n+1}\).
The best part is that the number of correct decimal places doubles after each iteration because it has second order convergence.

Answer 35

Oooh I see what you're doing now

Answer 36

Do you want me to get you started, or that's good enough?

Answer 37

Oh that's good enough thank you very much, because the original question just asked to mention if there is any roots if so state the number of them, so I was just curious to find the coordinate for it

Answer 38

thanks! @mathmate

Answer 39

If you are curious to use the method, you can check your answer :
x=0.03485461440500114

Answer 40

If you are curious to use the method, you can check your answer :
x=0.03485461440500114
accurate to the last digit, after just a few iterations.

Answer 41

lol that's not a few xD

Answer 42

you're welcome! :)
by the way, it took 4 iterations, starting from 0.03, the average of 0.01 and 0.05.

Answer 43

oh oops I made a mistake with the starting value

Answer 44

Exactly, the key is to find a good starting value! But any one reasonably close to the final root is ok, depending on the problem, and the conditions of convergence.

Answer 45

ah now I got it right
thnx!

Answer 46

Good, have fun with math! :)