What is a Logarithm?

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 = log4 = 64. Similarly, we know 10³ = 1000, then 3 = Log10 = 1000. 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.

Logarithms Rules

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

1. Product Rule

Logb (P × Q) = Logb P + logb Q

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

2. Quotient Rule

Logb (P/Q) = Logb P - logb Q

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

3. Power Rule

Logb (pq) = q. logb 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

Logb (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

Logb (by) = 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

b log b (y) = 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

Logy (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.

Logarithm Formulas

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

Basic Logarithm Formula

Some of the Different Basic Logarithm Formula are Given Below:

Logb (mn) = Logb (m) + Logb (n)

Logb (m/n) = Logb (m) - Logb (n)

Logb (xy) = y logb (x)

Logb \[\sqrt[m]{n}\] = Logb (n)/ m

m logb (x) + n logb (y) = logb ( xmy n )

Addition and Subtraction

Logb (m + n) = Logb m + Logb (1 + n/m)

Logb (m - n) = Logb m + Logb (1 - n/m)

Change of Base Formula

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

Lognm = logdm /logdn

Solved Examples

1. Solve the Following:

2log4 29

Solution:

Given,

2log4 29

Using change of base formula n we get

logbx = logdx / log104

2log4 29 = log1029/ log104

2log4 29 = 2 × 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:

26 = 6

Hence, 26 = 2 × 2 × 2× 2× 2× 2 = 64

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

Solution:

Log 5x + log (2x+3) = 1 + 2 log (3-x)

Log 5x + log (2x+3) = log10 + log (3-x) 2

Log (5x (2x+3) = 10(3-x)2)

5x (2x+3) = 10(3-x)2

10x2 + 15x = 10(9- 6x + x2)

10x2 + 15x = 90 - 60x +10x2

75x = 90

X = 90/75

X = 6/5

Quiz Time

1. Which of the Following Statements is Not True?

log7 7 = 1

log ( 4 + 3) = log ( 4 * 3)

log10 1 = 0

Log ( 4 + 5 + 6) = log 4 + log 5 + log 6

2. What is the Value of Log \[\sqrt{8}\] / Log 8 ?

1/\[\sqrt{8}\]

¼

½

1/8

FAQ (Frequently Asked Questions)

1. What are the Logarithms Used for?

Logarithms are used to calculate the potency of the earthquake.

Logarithms are used to determine the level of noise with respect to decibels, such as a sound made by a bell.

In Chemistry, logarithms are used to ascertain the acidity or pH level.

Lofartith is used to ascertain the monetary growth on a specific rate of interest.

Logarithms are widely used to ascertain the time required to decay or grow exponentially. For example, the growth of bacteria, radioactive decay, etc.

It is also used in mathematical calculations where multiplication changes into addition or vice versa.

In rule 1, the characteristic part of a logarithm is one less than the number of digits placed on the left side of the decimal point in the given number.

2. What are the Different Parts of a Logarithm?

There are two different parts of a logarithm. These are as follows:

1. Characteristic Part = The inner part of the logarithm of a number is known as the characteristic of a logarithm.

Rule 1: When the logarithm of a number is greater than 1.

Rule 2: When the logarithm of a number is less than 1.

In rule 1, the characteristic part of a logarithm is one less than the number of digits placed on the left side of the decimal point in the given number.

In rule 2, the characteristic part of a logarithm is negative and one more than the number of zeros placed between the decimal point and the first significant digit of the number. We can write 1̅, 2̅, rather than -1 or -2, etc

Number | Characteristic |

612. 25 | 2 |

16.291 | 1 |

2.1854 | 0 |

0.9413 | 1̅ |

0.03754 | 2̅ |

2. Mantissa Part - The decimal portion of the logarithm of a number is considered as the mantissa part of a number. The mantissa part of a number is usually determined from the log table.