# how to calculate log 3 and other examples of type 4 log, 5 log?

lfryerda | High School Teacher | (Level 2) Educator

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Your calculator or software such as Excel, Wolfram Alpha or Maple uses very sophisticated techniques for calculating logarithms of any value, such as `log 3` .  Using the multiplication and division properties of logarithms

`log (ab)=log a+log b`

`log(a/b)=log a-log b`

the change-of-base formula

`log_a b={log_x b}/{log_x a}`

and the Taylor series for the natural logarithm

`ln(1+x)=x-x^2/2+x^3/3-x^4/4+cdots`

it is possible to evaluate `ln x` at a finite number of points for x from 0 to 1 to an arbitrary decimal precision.  Then, using the scaling properties of the multiplication and division rules, you can get any value of `ln x`.  Finally, the change-of-base formula is used to calculate a logarithm in some other base, such as 10, 2 or 7.

The calculator also use interpolation techniques, improved identities or techniques such as Newton's method over the Taylor series for faster calculation and control of rounding errors.

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