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Logarithms and Exponents

Logarithms

The logarithm of any number is the exponent of the power to which 10 must be raised to produce the number. The logarithm X of the number N to the base 10 is the exponent of the power to which 10 must be raised to give N (for example, log₁₀ N = X). Logarithms consist of two parts. First, there is the “characteristic,” which is determined by the position of the first significant figure of the number in relation to the decimal point. If we count leftwards from the decimal point as positive and rightwards as negative, the characteristic is equal to the count ending at the right of the first significant figure. Thus, the characteristic of the logarithm of 2340 is 3, and of 0.00234 is −3. Second, there is the “mantissa.” It is always positive, is found in logarithm tables, and depends only on the sequence of significant figures. Thus, the mantissa for the two numbers is the same, namely 0.3692. The logarithm of a number is the sum of the characteristic and the mantissa. Thus, log 2340 = 3.3692 while log 0.00234 = −3 + 0.3692 = −2.6308.

The logarithms of the whole integers 1 to 10 are given below.

Useful Rules in Handling Logarithms

1. The logarithm of a product is equal to the sum of the logarithm of the factors:
2. The logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator:
3. How about log 2.5? The answer from the log tables is 0.398.
4. The logarithm of the reciprocal of a number is the negative logarithm of the number:
5. The logarithm of a number raised to a power is the logarithm of the number multiplied by the power:

Exponents

It is convenient to express large numbers as 10^x, where x represents the number of places that the decimal must be moved to place it after the first significant figure. This also represents 10 • 10 for x times. For example, 1,000,000 may be expressed as 1 × 10⁶; 3663 as 3.663 × 10³; and so on. To multiply, the exponents are added, but coefficients are multiplied. To divide, the exponents are subtracted but coefficients are divided.

Multiplying: (1 × 10^x) • (1 × 10^y) = 1 × 10^{x + y} (4 × 10²) • (2 × 10³) = 8 × 10⁵

Dividing: (1 × 10^x ÷ (1 × 10^y) = 1 × 10^{x − y} (4 × 10²) ÷ (2 × 10³) = 2 × 10⁻¹

Numbers less than 1 are 10^−x. For example, 0.000001 is 1 × 10⁻⁶.

Multiplying: (1 × 10^−x) • (1 × 10^−y) = 1 × 10^{−(x + y}) (4 × 10⁻²) • (2 × 10⁻³) = 8 × 10⁻⁵

A large number multiplied by a small number: (4 × 10⁻²)(2 × 10³) = 8 × 10¹

(Logarithms and Exponents are reproduced through the courtesy of Dr. Richard B. Brandt, Dept. of Biochemistry, MCV, VCU, Richmond, Virginia, 23298).

Reference Tables for Chemistry