Dyalog APL (no builtin), 13 12 bytes
Dyalog does have the builtin ⍟, but this way it's more fun :)
(⊢-1-÷∘*)⍣=⍨
Basically just Newton's method:
$$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$
where (L is the number we are taking the log of)
$$ f(x) = e^x - L \\ f'(x) = e^x \\ x_{n+1} = x_n - \frac{e^{x_n} - L}{e^{x_n}} = x_n - 1 - \frac L {e^{x_n}} $$
(⊢-1-÷∘*)⍣=⍨
⍨ ⍝ x_0 is L
⍣= ⍝ repeat Newton's method until x_n and x_n+1 are "equal":
⊢-1- ⍝ x_n - 1 -
÷∘* ⍝ L / e^x_n
💎 Created with the help of Luminespire at https://vyxal.github.io/Luminespire
"Equal" here means
$$ \left|x_n-x_{n+1}\right|\leq10^{-14}\max\left(\left|x_n\right|, \left|x_{n+1}\right|\right) $$
| x | ln(x) | builtin ln(x) | steps until convergence |
|---|---|---|---|
| 0.1 | -2.3025850929940455E0 | -2.3025850929940455 | 8 |
| 0.25 | -1.3862943611198906E0 | -1.3862943611198906 | 7 |
| 0.5 | -6.9314718055994520E-1 | -0.6931471805599453 | 7 |
| 0.75 | -2.8768207245178090E-1 | -0.2876820724517809 | 7 |
| 0.9 | -1.0536051565782634E-1 | -0.10536051565782628 | 7 |
| 1 | 1.1102230246251565E-16 | 0 | 8 |
| 1.3 | 2.6236426446749117E-1 | 0.26236426446749106 | 7 |
| 2 | 6.9314718055994530E-1 | 0.6931471805599453 | 7 |
| 2.718281828459045 | 9.9999999999999990E-1 | 1 | 8 |
| 3.141592653589793 | 1.1447298858494002E0 | 1.1447298858494002 | 8 |
| 4 | 1.3862943611198906E0 | 1.3862943611198906 | 9 |
| 5 | 1.6094379124341003E0 | 1.6094379124341003 | 9 |
| 7 | 1.9459101490553135E0 | 1.9459101490553132 | 11 |
| 10 | 2.3025850929940460E0 | 2.302585092994046 | 14 |
| 53 | 3.9702919135521220E0 | 3.970291913552122 | 55 |
| 54.59815003314423 | 4 | 4 | 57 |
| 99 | 4.5951198501345900E0 | 4.59511985013459 | 100 |
gotta say, ln(1) resulting in 1e-16 is kinda annoying but well within the spec
This is 1 byte less than an implementation of Jos Woolley's, 1e9ׯ1+*∘1e¯9, but probably still has some room to be golfed further.