Acevedo algorithm

1. Acevedo’s Algorithm
The Acevedo's Algorithm is named after José Acevedo Jiménez
who developed it. This is expressed as follows:
𝐴1 =
𝑎
𝑏 𝑥0
𝐴2 =
(
𝑎
𝑏 𝑥0
)
𝐵
𝑏 𝑥1
𝐴3 =
(
(
𝑎
𝑏 𝑥0
)
𝐵
𝑏 𝑥1
)
𝐵
𝑏 𝑥2
𝐴 𝑛+1 = ((
(
(
𝑎
𝑏 𝑥0
)
𝐵
𝑏 𝑥1
)
𝑏 𝑥2
⋮
𝐵
)
⋰
)
𝐵
𝑏 𝑥 𝑛

2. if
1 ≤ 𝐴 𝑛+1 < 𝑏
then
log 𝑏 𝑎 = 𝑥0 + ∑ 𝑥𝑖 𝐵−𝑖
𝑛
𝑖=1
Where:
𝑎 ∈ ℝ ≥ 1.
𝐵 = number system base, 𝐵 > 1.
𝑏 = logarithm base.
Original sequence of the algorithm developed by Acevedo to calculate logarithms.

3. Example: Calculate the logarithm of seven in base 10 with three decimal
places.
𝑎 = 7, 𝑏 = 10, 𝐵 = 10
(
(
(
(
(
7
100)
10
108 )
10
104
)
105
10
)
10
100
)
10
109
< 7
log 𝑎 ≈ 𝑥0 +
𝑥1
𝐵−1
+
𝑥2
𝐵−2
+
𝑥3
𝐵−3
+
𝑥4
𝐵−4
+
𝑥5
𝐵−5
log10 7 ≈ 0 +
8
101
+
4
102
+
5
103
+
0
104
+
9
105
log 7 ≈ 0.84509