2.9 Logarithmic Expressions

log_b(c) = a means exactly the same thing as bᵃ = c. Here b is the base, with b > 0 and b ≠ 1.

So log₂(8) = 3 because 2³ = 8, and log₅(1/25) = −2 because 5⁻² = 1/25.

Because bᵃ is always positive, you can only take the log of a positive number. log₂(0) and log₂(−4) are undefined.

When no base is written, the base is 10: log(1000) = 3 because 10³ = 1000. This is the common logarithm.

The natural logarithm uses base e ≈ 2.718 and is written ln: ln(x) = log_e(x). So ln(e⁴) = 4 and ln(1) = 0.

Calculators have a LOG key for base 10 and an LN key for base e. For any other base you need the change-of-base rule from topic 2.12, or a logBASE command.

Some values come straight from arithmetic:

• log_b(1) = 0 and log_b(b) = 1 for any valid base.
• log_b(bⁿ) = n: log₃(81) = 4 since 81 = 3⁴.
• Fractions give negative logs: log₂(1/16) = −4.
• Roots give fractional logs: log₄(2) = 1/2 since 4^(1/2) = 2, and log₄(8) = 3/2 since 4^(3/2) = 8.

For other values, trap the number between powers of the base. log₂(50) is between 5 and 6, because 2⁵ = 32 and 2⁶ = 64. A calculator gives log₂(50) ≈ 5.644.

Logs grow slowly. log(1,000,000) is only 6. Multiplying the input by 10 adds just 1 to a common log.

Known values help too. Since 10^0.301 ≈ 2, you also know 10^1.301 ≈ 20 and 10^2.301 ≈ 200. So log 20 ≈ 1.301 and log 200 ≈ 2.301. Each extra factor of 10 adds exactly 1 to the common log.

On a regular (linear) scale, equally spaced marks go up by the same amount: 0, 1, 2, 3. On a logarithmic scale, equally spaced marks go up by the same factor: with base 10, the marks are 10⁰, 10¹, 10², 10³, that is 1, 10, 100, 1000.

Each unit of distance on a log scale stands for multiplying by the base. That's how scales like pH and decibels can cover huge ranges of values in a small space.

To place a value on a base-10 log scale, use its common log as its position. The value 500 sits at log 500 ≈ 2.699, about 70% of the way from the 100 mark (position 2) to the 1,000 mark (position 3), not halfway.