Question

How do you calculate `log_10 (7)`?

Answer 1

You can use Newton's method to find approximations...

Explanation:

`log_10(7)` is an irrational number with no simpler representation.

Here's one way to find numerical approximations for it without the benefit of a `ln` or `log` function...

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Newton's method

Define `f(x) = 10^x-7`

Then `f'(x) = 10^x*ln(10)`

Starting with approximation `a_0 = 1`, use Newton's method, iterating using the formula:

`a_(i+1) = a_i - (f(a_i))/(f'(a_i)) = a_i - (10^(a_i) - 7)/(10^(a_i)*ln(10))`

Of course this requires that you are able to calculate `10^x` and know a reasonable approximation for `ln(10)` (say `2.3026`).

For example, if we use `ln(10) ~~ 2.3026` then the iterates look like this:

`1.00000000000000`
`0.86971249891427`
`0.84578273695874`
`0.84509858389730`
`0.84509804001812`
`0.84509804001426`
`0.84509804001426`

Note that we do not need to know `ln(10)` very accurately in order to find `log_10(7)` - it just affects the rate of convergence.