Can log_2(x) be expressed as a series?

  • Thread starter geor
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In summary, the series for log<sub>2</sub>(x) is the sum of the infinite terms 1/n * (x-1)<sup>n</sup>, derived from the Taylor series expansion for ln(x) with x substituted as (x-1). It converges for all positive real numbers and its accuracy depends on the number of terms used. It cannot be used to calculate log<sub>2</sub>(x) for negative numbers.
  • #1
geor
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Hello all,

I am aware that we can write ln(x) as a series.
But what can we say for a logarithm of an arbitrary base?

Can we write for example log_2(x) as series?

Thanks in advance..
 
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  • #2
Remember that the logarithm in base b of a number is just a constant multiple of the natural logarithm of the number, specifically: logb(x) = ln(x)/ln(b).
 
  • #3
Oh yes, of course! Thanks a lot..
 

1. What is the series for log2(x)?

The series for log2(x) is the sum of the infinite terms in the form of 1/n * (x-1)n, where n ranges from 1 to infinity.

2. How do you derive the series for log2(x)?

The series for log2(x) can be derived by using the Taylor series expansion for ln(x) and then substituting x with (x-1) since log2(x) = ln(x) / ln(2).

3. What is the convergence of the series for log2(x)?

The series for log2(x) converges for all values of x greater than 0. This means that the series is a valid representation of log2(x) for all positive real numbers.

4. How accurate is the series for log2(x)?

The accuracy of the series for log2(x) depends on the number of terms used in the calculation. The more terms that are included, the more accurate the approximation will be.

5. Can the series for log2(x) be used to calculate log2(x) for negative numbers?

No, the series for log2(x) is only valid for positive real numbers. To calculate log2(x) for negative numbers, a different series or method must be used.

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