Great question—let’s break this down step by step to clear up that confusion!

Your Initial Misconception: Logarithm Bases vs. Exponential Bases #

You’re right that the base of a logarithm can match the base of an exponential (like log₂(2⁷⁵) = 75), but logarithms are far more flexible than that—this is exactly why the change of base formula exists. The core idea of a logarithm is simple:

log_b(x) answers the question: “What power do I need to raise b to, to get x?”

The base b is just a reference scale for this question. The change of base formula (log_b(x) = log_k(x)/log_k(b) for any positive k ≠ 1) lets us translate this question between different scales. That’s why we can use any base we want—we’re just converting from one “power scale” to another.

Why Base-10 Logarithms Work for Counting Decimal Digits #

Let’s dive into the 2⁷⁵ digit-counting example, since this is where base 10 becomes essential:

• In the decimal system (base 10), every number with n digits falls in the range:
  10^(n-1) ≤ x < 10^n
  For example, 3-digit numbers are from 100 (10²) up to 999 (which is less than 10³).
• To find how many digits 2⁷⁵ has, we need to find the smallest integer n where this inequality holds for x = 2⁷⁵.
• Take log₁₀ of all parts of the inequality to simplify it:
  n-1 ≤ log₁₀(2⁷⁵) < n
• Using the logarithm power rule (log_b(a^c) = c*log_b(a)), we rewrite this as:
  n-1 ≤ 75*log₁₀(2) < n
• We know log₁₀(2) ≈ 0.3010, so 75*0.3010 ≈ 22.575.
• This means n-1 ≤ 22.575 < n, so n-1 is the floor of 22.575 (22), and n = 22 + 1 = 23 digits. That’s exactly the calculation you showed!

Your Hunch is Correct: Logarithm Bases and Number Systems #

You guessed it—logarithm bases directly align with number systems:

• Base 10: Ties to decimal, as we saw. It tells us how many digits a number has in decimal (using floor(log₁₀(x)) + 1).
• Base 2: Aligns with binary. floor(log₂(x)) + 1 gives the number of bits needed to represent x in binary. For example, 8 is 1000 in binary—floor(log₂(8)) +1 = 3 +1 =4 bits, which matches.
• Base 16: Yes, this corresponds to hexadecimal! The number of hex digits for x is floor(log₁₆(x)) +1. Since log₁₆(x) = log₂(x)/4, this makes sense because each hex digit represents 4 binary bits.

Final Takeaways #

• Logarithm bases are just reference scales for the question: “What power gives this number?”
• We use base 10 for digit counts because our everyday number system is base 10—its powers define the ranges of digit lengths.
• Your intuition about bases matching number systems is spot-on; each base’s logarithm directly relates to how we represent numbers in that base.

内容的提问来源于stack exchange，提问作者Rijndael