Logarithm Formula

logarithm formula

The logarithm is indeed an exponential or power that must be applied to a base in order to achieve a particular number. Logarithms are represented mathematically as m is indeed the Logarithm of n towards the base b if b^m = n, which may alternatively be written as m = log_b n. For example,10³ = 1000, then 3 = log₁₀1000

We also know that 10³ = 1000, hence 3 = log₁₀1000. Common or Briggsian logarithms are sometimes referred to as common and Briggsian logarithms and are simply represented as log n.

Rules of Logarithms

There are seven Logarithm principles that are important for expanding, contracting, and solving Logarithmic equations. The following are the seven Logarithms rules:

1. Product Rule

Log_b(P×Q) = log_bp + log_bQ

The product’s logarithm is equal to the sum of the components’ logarithms.

2. Rule of the Quotient

Log_b(P/Q) = log_bP − log_bQ

The difference between the numerator and denominator logarithms is the logarithm of the ratio of two integers.

Basic Logarithm Formula

Some of the Different Basic Logarithm Formula are Given Below:

Examples

Example 1: a) 5³ = 125 b) 3^-3 = 1 / 27.

Solution:

Using the definition of the logarithm,

B^x = a ⇒ logb_b a = x

Using this,

a) 5³ = 125 ⇒ log₅ 125 = 3

b) 3^-3 = 1 / 27 ⇒ log₃ 1/27 = -3

Answer: a) log₅ 125 = 3; b) log₃ 1/27 = -3.

Example 2: 5 log x + log y – 8 log z.

Solution:

To find: The compressed form of the given expression as a single logarithm using logarithm formulas.

5 log x + log y – 8 log z

= (5 log x – 8 log z) + log y (Regrouped the terms)

= (log x⁵ – log z⁸) + log y (∵ a log x = log x^a)

= log (x⁵/z⁸) + log y (∵ log x – log y = log (x/y)

= log (x⁵y/z⁸) (∵ log x + log y = log (xy)

Answer: 5 log x + log y – 8 log z = log (x⁵y/z⁸).

Example 3: Find the integer value of log3 (1/9) using log formulas.

Solution:

Log₃ (1/9) = log₃ 1 – log₃ 9 (∵ log_b (x / y) = log_b x – log_b y)

= 0 – log₃ 32 (∵ log_b 1 = 0)

= – 2 log₃ 3 (∵ log_b ax = x log_b a)

= -2 (1) (∵ log_b b = 1)

= -2

Answer: log₃ (1/9) = -2.