Laws of logarithm

These are similar to all the laws of indices.

It shows clearly that a logarithm is a mirror image of an index.

e.g.  100 = 10²; 2 = log₁₀100

1000 = 10³; 3 = log₁₀1000

0.01 = 10^-2; -2 = log₁₀^0.01

Evaluate the following:

(1) If

1. Product

Given that 100 = 100²  log₁₀ 100 = 2 and 1000 = 10³  log₁₀ 1000 = 3 then

100 x 1000 = 10² x 10³ = 10²⁺³ = 10⁵ (1^st law, indices)

log₁₀(100 x 1000) = 5 which is equal to 2+3

= log₁₀ 100 + log₁₀ 1000

Example

Given that log₁₀ 2 = 0.0310, log₁₀ 3 = 0.4771 and log₁₀ 7 = 0.8451, evaluate

Log₁₀ 42

Log₁₀ 42 = log₁₀ (7 x 6) = log₁₀ (7 x 2 x3)

= 0.8451 + 0.0310 + 0.4771 = 1.6232.

The logarithm of a product is the sum of the logarithms of the factors that make up the product.

2. Quotient

Since 1000 ¸ 100 = 10³ ¸ 10² = 10^3-2 = 10¹

log₁₀(1000 ¸ 100) = 1 = log₁₀(1000) – log₁₀(100). This is the 2^nd law of indices.

Example

Log₁₀(14/3)

Log₁₀(14/3) = log₁₀ 14– log₁₀ 3

= log₁₀ (2 x 7) – log₁₀ 3

= log₁₀ 2 + log₁₀ 7 – log₁₀ 3

= (0.3010 + 0.8451) – 0.4771

= 0.6690

The logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.

3. Raising to a Power

Example

Log₁₀(100)² = 2log₁₀ 100 = 2 x 2

Evaluate log₁₀ 8, if log₁₀ 2 = 0.3010

log₁₀ 8   = log₁₀ 2³ = 3log₁₀ 2

= 3 x 0.3010

= 0.9030.

4. Roots

The logarithms of the n^th root of a number is the logarithm of the number, divided by n

e.g. ³  1000 = ³ 10³ = ³  10 x 10 x 10 = 10 or 10¹

log ³  1000 = 1 or (log 1000) ¸ 3 = 1

log_b ⁿ  x = 1/n log_b X.