If nnn numbers are in arithmetic progression (AP), their logarithms will not generally form an arithmetic progression. Instead, the logarithms of numbers in an arithmetic progression form a sequence that reflects a different pattern. Key insight:

• If the numbers themselves form an arithmetic progression, i.e., the terms are a,a+d,a+2d,…a,a+d,a+2d,\ldots a,a+d,a+2d,…, then their logarithms are:

log⁡(a),log⁡(a+d),log⁡(a+2d),…\log(a),\log(a+d),\log(a+2d),\ldots log(a),log(a+d),log(a+2d),…

This sequence is generally not arithmetic because:

log⁡(a+(k+1)d)−log⁡(a+kd)≠constant\log(a+(k+1)d)-\log(a+kd)\neq \text{constant}log(a+(k+1)d)−log(a+kd)=constant

The difference between consecutive logarithms depends on the ratio:

log⁡(a+(k+1)d)−log⁡(a+kd)=log⁡(a+(k+1)da+kd)\log(a+(k+1)d)-\log(a+kd)=\log\left(\frac{a+(k+1)d}{a+kd}\right)log(a+(k+1)d)−log(a+kd)=log(a+kda+(k+1)d​)

which changes with kkk.

• However, if the logarithms themselves form an arithmetic progression, then the original numbers form a geometric progression. This is because an arithmetic progression in the logarithms means:

log⁡xk+1−log⁡xk=d=constant\log x_{k+1}-\log x_k=d=\text{constant}logxk+1​−logxk​=d=constant

which implies:

xk+1xk=constant ratio\frac{x_{k+1}}{x_k}=\text{constant ratio}xk​xk+1​​=constant ratio

i.e., xkx_kxk​ is geometric.

Summary:

• Numbers in arithmetic progression ⇒\Rightarrow ⇒ logarithms form a sequence with non-constant differences (not an AP).
• Logarithms in arithmetic progression ⇒\Rightarrow ⇒ numbers form a geometric progression.

This is supported by the example and explanation in the search results where logarithms of numbers in arithmetic progression do not form an arithmetic progression themselves, but the logarithms forming an AP imply the original numbers are in geometric progression

. Hence, the logarithm of numbers in arithmetic progression will vary in a manner that is not arithmetic; the differences between consecutive logarithms decrease as the terms increase, reflecting a concave pattern.