Log/Ln: Find Two Term Approximation

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In summary, the conversation is about finding a two term approximation for a function near a specific value. The function in question is ln(x+1) and the method being used is a power series expansion. The conversation also includes the use of a hint and substitution to simplify the problem.
  • #1
Physics197
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If you had something like ln(x+1) is there are way of breaking that term up?

I have a question where I have to find a two term approximation and I'm at a stage similar to this one and I don't know what I can do.
 
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  • #2
What do you mean by breaking it up? There is a power series for ln(1+x)

x-x2/2+x3/3+...
 
  • #3
I have y=ln(3u+4) and need a 2 term approximation (we are given a hint 4 = 3+1) so I assume that we must do: y=ln(3u+1+3) and the only way we know to approximate is y=ln(1+u) ~ u so in order to get that I need to somehow get rid of the 3 in y=ln(3u+1+3)
 
  • #4
What kind of approximation.
We know log(x+1)~x
so expansion about x+1=1->x=0 would be nice
let f(x)=log(x+1)
f(x)~f(0)+f'(0)x+f''(0)x^2/2+f''(0)x^3/6+...
 
  • #5
The entire question is:

Find a two term approximation for the following function near t=1.

h = 5ln(3t+1)
and after substituting u+1 for t

I get:

h = 5ln(3u+4)
Now it gives us the hint: 4 = 3+1

But I am stuck after this.
 
  • #6
Physics197 said:
The entire question is:

Find a two term approximation for the following function near t=1.

h = 5ln(3t+1)
and after substituting u+1 for t

I get:

h = 5ln(3u+4)
Now it gives us the hint: 4 = 3+1

But I am stuck after this.
Let v=3u+1, so h=5ln(1+v)~5[v - v2/2 + ...]
Take it from there.
 

1. What is the purpose of finding a two-term approximation for Log/Ln?

The purpose of finding a two-term approximation for Log/Ln is to simplify complex logarithmic expressions into a more manageable form. This approximation can also help in estimating values for logarithmic functions and making calculations easier.

2. How do you find the two-term approximation for Log/Ln?

To find the two-term approximation for Log/Ln, you can use the Maclaurin series expansion formula, which is: Log/Ln(x) = (x-1) - (x-1)^2/2. This formula can be used to approximate the natural logarithm of any number close to 1.

3. Can the two-term approximation for Log/Ln be used for all logarithmic functions?

No, the two-term approximation for Log/Ln is only accurate for values close to 1. For other values, higher order approximations may be needed for better accuracy.

4. What are the limitations of using the two-term approximation for Log/Ln?

The two-term approximation for Log/Ln is only accurate for values close to 1 and cannot be used for negative values. It also does not take into account any logarithmic rules or properties, which may be important in some calculations.

5. How can the two-term approximation for Log/Ln be used in real-world applications?

The two-term approximation for Log/Ln can be useful in various fields such as physics, engineering, and finance. It can help in simplifying complex equations, estimating values, and making calculations easier. For example, it can be used in calculating compound interest rates or in modeling exponential decay in radioactive materials.

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