high school Common Logarithm

Common Logarithm

A common logarithm uses base 10, splitting any positive number into an integer characteristic and a fractional mantissa between 0 and 1. The characteristic counts the digits and the mantissa fixes the pattern, so the length of a huge number like 2 to the 30th falls out without multiplying it. Each time x grows tenfold the log value rises by 1, and that rule is the digit count. Move x and the log value splits into characteristic and mantissa.

Logarithm Base 10

Argument x200

👀 See It

①The common logarithm has base 10: log x = log_{10} x

②The common log of a positive number splits into an integer part (characteristic) and a fractional part in [0,1) (mantissa)

③Each time x grows tenfold, the log value increases by 1

④This sketch uses N ≥ 2. If 0 < N < 1 the characteristic is negative

Characteristic and Mantissa

Characteristic + mantissa

log N = (characteristic) + (mantissa), 0 ≤ mantissa < 1

The characteristic is an integer, the mantissa is in [0,1) — they hold the digit count and the digit pattern separately

Digit count and characteristic

integer N has d digits ⇔ characteristic = d − 1 ⇔ 10^d-1 ≤ N < 10^d

If the characteristic is d−1, then N is a d-digit integer

Equal Mantissas Mean Equal Digit Patterns

🔢 Shift by a Power of 10

①Two numbers with the same mantissa share the digit pattern and differ only by a power of 10

②e.g., log 2 and log 200 have the same mantissa

③The characteristic sets the decimal position (digit count); the mantissa sets the digits

Compute It Directly

Example 1

Given log 2 = 0.3010, find log 200.

Factor 200 = 2 × 100 = 2 × 10².

log 200 = log 2 + log 10²

Add log 10² = 2.

= 0.3010 + 2 = 2.3010

▸ 2.3010 (char. 2, mant. 0.3010)

A power of 10 changes only the characteristic, leaving the mantissa (0.3010) intact.

Example 2

How many digits does 2³⁰ have? (log 2 = 0.3010)

Take the common logarithm.

log 2³⁰ = 30 log 2 = 30 × 0.3010 = 9.03

The characteristic is 9, so the digit count is 9 + 1.

char. 9 ⇒ 10 digits

▸ 10 digits

The digit count of a power comes straight from the log characteristic + 1.

Wrap-up

Key result

log N = char. + mant., d digits ⇔ char. = d − 1

Characteristic = digit count, mantissa = digit pattern — separating them is the key

exam-style

How many digits does 3²⁰ have? (log 3 = 0.4771)

①9 digits

②10 digits

③11 digits

④12 digits

⑤13 digits

▸ ② 10 digits

Take the common logarithm to get the characteristic.

log 3²⁰ = 20 × 0.4771 = 9.542

The characteristic is 9, so digit count = 9 + 1.

char. 9 ⇒ 10 digits

🎯 Exam Points

①log N = char. + mant. (0 ≤ mant. < 1)

②d-digit integer ⇔ char. = d − 1

③digit count of a power = log characteristic + 1

④equal mantissas ⇒ same digit pattern

⑤if 0<N<1 the characteristic is negative (position of first nonzero decimal)

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Logarithmic Function

Trigonometric Functions

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