Logarithm Formula

The Logarithm is an exponent or power  to which a base must be raised to obtain a given number. Mathematically, Logarithms are expressed as, m is the Logarithm of n to the base b if bm = n, which can also be written as m = logb n. For example, 43 = 64; hence 3 is the Logarithm of 64 to base 4, or 3 = log464. Similarly, we know 103 = 1000, then 3 = log101000. Logarithms with base 10 are usually known as common or Briggsian Logarithms and are simply expressed as log n. In this article, we will discuss what is a Logarithm, Logarithms formulas, basic Logarithm formulas, change of base rule, Logarithms rules and formulas, what is Logarithm used for etc.

There are 7 Logarithm rules which are useful in expanding Logarithm, contracting Logarithms, and solving Logarithmic equations. The seven rules of Logarithms are discussed below:

1. Product Rule

\[\log_{b}{(P \times Q)} = \log_{b}{P} + \log_{b}{Q}\]

The Logarithm of the product is the total of the Logarithm of the factors.

2. Quotient Rule

\[\log_{b}{(\frac{P}{Q})} = \log_{b}{P}  -  \log_{b}{Q}\]

The Logarithm of the ratio of two numbers is the difference between the Logarithm of the numerator and denominator. 

3. Power Rule

\[\log_{b}{(P^{Q})} = q \times  \log_{b}{P} \]

The above property of the product rule states that the Logarithm of a positive number p to the power q is equivalent to the product of q and log of p.

4. Zero Rule  

\[\log_{b}{ (1)} = 0 \]

The Logarithm of 1 such that b greater than 0 but b≠1, equals zero. 

5. The Logarithm of a Base to a Power Rule

\[\log _{b}{b^{y}} = y \]

The Logarithm of a base to a power rule states that the Logarithm of b with a rational exponent is equal to the exponent times its Logarithm.

6. A Base to a Logarithm Power Rule 

blogy = y

The above rule states that raising the Logarithm of a number to the base of a Logarithm is equal to the number.

7. Identity Rule

\[\log_{y}{y} = 1 \]

The argument of the Logarithm (inside the parentheses) is similar to the base. As the base is equal to the argument, y can be greater than 0 but cannot be equals to 0.

Below are some of the different Logarithm formulas which help to solve the Logarithm equations.

Some of the Different Basic Logarithm Formula are Given Below:

\[\log_{b}{(m \times n)} = \log_{b}{m} + \log_{b}{m}\]

\[\log_{b}{(\frac{m}{n})} = \log_{b}{m}  -  \log_{b}{n}\]

\[\log_{b}{(x^{y})} = y \times  \log_{b}{x} \]

\[\log_{b}{\sqrt[m]{n}} = \log_{b}{n^{\frac{1}{m}}} \]

\[m \log_{b}{(x)} + n \log_{b}{(y)} = \log_{b}{(x^{m}y^{n})} \]

\[\log_{b}{(m + n)} = \log_{b}{m} + \log_{b}{(1 + \frac{n}{m})} \]

\[\log_{b}{(m - n)} = \log_{b}{m} + \log_{b}{(1 - \frac{n}{m})} \]

In the change of base formula, we will convert the Logarithm from a given base ‘n’ to base ‘d’.

\[\log_{n}{m} = \frac{\log_{d}{m}}{\log_{d}{n}}\]

1. Solve the Following:  2 log429

Solution:

Given,

 \[2 \log_{4}{29} \]

Using change of base formula n we get

\[\log_{b}{x} = \frac{\log_{d}{m}}{\log_{10}{4}}\]

\[2 \log_{4}{ 29} = \frac{\log_{10}{29}}{\log_{10}{4}}\]

\[2 \log_{4}{ 29} = 2 \times 2.43\]

= 4.86

2. Find the Value of x in log2x = 6

Solution:

The Logarithm function given above can be expressed in the exponential form as:

\[2^{6} = 64 \]

Hence, \[2^{6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \]

3. Find log 5x + log (2x+3) = 1 +  2 log (3-x) , when x