## 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,10^{3} = 1000, then 3 = log_{10}1000

We also know that 10^{3} = 1000, hence 3 = log_{10}1000. 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_{b}p + log_{b}Q

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

### 2. Rule of the Quotient

Log_{b}(P/Q) = log_{b}P − log_{b}Q

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) 5 ^{3} = 125 b) 3^{-3} = 1 / 27.**

**Solution:**

Using the definition of the logarithm,

B^{x} = a ⇒ logb_{b} a = x

Using this,

a) 5^{3} = 125 ⇒ log_{5} 125 = 3

b) 3^{-3} = 1 / 27 ⇒ log_{3} 1/27 = -3

**Answer:** a) log_{5} 125 = 3; b) log_{3} 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 x^{5} – log z^{8}) + log y (∵ a log x = log x^{a})

= log (x^{5}/z^{8}) + log y (∵ log x – log y = log (x/y)

= log (x^{5}y/z^{8}) (∵ log x + log y = log (xy)

**Answer:** 5 log x + log y – 8 log z = log (x^{5}y/z^{8}).

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

**Solution:**

Log_{3} (1/9) = log_{3} 1 – log_{3} 9 (∵ log_{b} (x / y) = log_{b} x – log_{b} y)

= 0 – log_{3} 32 (∵ log_{b} 1 = 0)

= – 2 log_{3} 3 (∵ log_{b} ax = x log_{b} a)

= -2 (1) (∵ log_{b} b = 1)

= -2

**Answer:** log_{3} (1/9) = -2.