If nnn numbers are in arithmetic progression, then their logarithms (to any fixed base) form neither an arithmetic progression nor a geometric progression in general; they follow a concave, slowly increasing pattern given by the log function applied to a linear sequence.

Core idea (short answer)

Let the nnn numbers in arithmetic progression be

a, a+d, a+2d, …, a+(n−1)d,(a>0, a+d>0,… )a,;a+d,;a+2d,;\dots,;a+(n-1)d,\quad (a>0,;a+d>0,\dots)a,a+d,a+2d,…,a+(n−1)d,(a>0,a+d>0,…)

Then their logarithms (say base 10 or base eee) are

log⁡(a), log⁡(a+d), log⁡(a+2d), …, log⁡(a+(n−1)d).\log(a),;\log(a+d),;\log(a+2d),;\dots,;\log(a+(n-1)d).log(a),log(a+d),log(a+2d),…,log(a+(n−1)d).

This is simply the logarithm of a linear function , which is a concave curve: the “steps” between successive terms keep decreasing.

So:

  • The original numbers grow in equal additive steps ddd (arithmetic progression).
  • Their logs grow in unequal , steadily shrinking steps; the sequence of logs is not an arithmetic progression, but has decreasing first differences.

Mini walkthrough: what “manner” do logs vary in?

Imagine the line

xk=a+(k−1)d,k=1,2,…,nx_k=a+(k-1)d,\quad k=1,2,\dots,nxk​=a+(k−1)d,k=1,2,…,n

This is a straight line in kkk.

Now apply logarithm:

yk=log⁡(xk)=log⁡(a+(k−1)d).y_k=\log(x_k)=\log\big(a+(k-1)d\big).yk​=log(xk​)=log(a+(k−1)d).

Key properties of log⁡(x)\log(x)log(x):

  • It is increasing: bigger xxx gives bigger log⁡x\log xlogx.
  • It is concave: its slope ddxlog⁡x=1x\frac{d}{dx}\log x=\frac{1}{x}dxd​logx=x1​ decreases as xxx increases.

Because of concavity:

  • The increments

yk+1−yk=log⁡(a+kd)−log⁡(a+(k−1)d)y_{k+1}-y_k=\log(a+kd)-\log(a+(k-1)d)yk+1​−yk​=log(a+kd)−log(a+(k−1)d)

get smaller as kkk increases.

  • So the logarithms rise, but each step is smaller than the previous one.

That is the “manner” of variation: increasing but with diminishing increments.

Intuitive picture

Think of an arithmetic progression like 10, 20, 30, 40, 50.
Their base‑10 logs are approximately:

  • log⁡10≈1\log 10\approx 1log10≈1
  • log⁡20≈1.301\log 20\approx 1.301log20≈1.301 (jump ~0.301)
  • log⁡30≈1.477\log 30\approx 1.477log30≈1.477 (jump ~0.176)
  • log⁡40≈1.602\log 40\approx 1.602log40≈1.602 (jump ~0.125)
  • log⁡50≈1.699\log 50\approx 1.699log50≈1.699 (jump ~0.097)

The jumps keep decreasing, illustrating the general rule: equal additive steps in the original numbers correspond to shrinking additive steps in their logarithms.

One more angle: relation to geometric progressions

Historically, logarithms were defined so that an arithmetic progression in the logarithms corresponds to a geometric progression in the original numbers.

  • If numbers are in geometric progression, their logs form an arithmetic progression.
  • Here we are in the opposite situation: numbers are in arithmetic progression, so their logs form neither an arithmetic nor a geometric progression in general; they trace out the logarithm of a straight line (a concave curve).

Tiny HTML table for clarity

Here is a small numeric example in HTML table form, matching your formatting rule:

html

<table>
  <tr>
    <th>Term index k</th>
    <th>AP term x_k</th>
    <th>log<sub>10</sub>(x_k)</th>
    <th>Difference in logs</th>
  </tr>
  <tr>
    <td>1</td>
    <td>10</td>
    <td>1.000</td>
    <td>–</td>
  </tr>
  <tr>
    <td>2</td>
    <td>20</td>
    <td>1.301</td>
    <td>0.301</td>
  </tr>
  <tr>
    <td>3</td>
    <td>30</td>
    <td>1.477</td>
    <td>0.176</td>
  </tr>
  <tr>
    <td>4</td>
    <td>40</td>
    <td>1.602</td>
    <td>0.125</td>
  </tr>
  <tr>
    <td>5</td>
    <td>50</td>
    <td>1.699</td>
    <td>0.097</td>
  </tr>
</table>

This table concretely shows that equal steps in the AP do not give equal steps in the logarithms; the log sequence has decreasing differences.

TL;DR:
If nnn numbers are in arithmetic progression, their logarithms form an increasing sequence with decreasing successive differences , i.e., they follow the concave curve y=log⁡(a+(k−1)d)y=\log(a+(k-1)d)y=log(a+(k−1)d), not a standard progression.

Information gathered from public forums or data available on the internet and portrayed here.