# Logarithm Formula

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## 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 = 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.

### Logarithms Rules

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

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

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

7. Identity Rulelogyy=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:

The Logarithm function given above can be expressed in the exponential form as:26=64

Hence, 26=2×2×2×2×2×2=64

75x=90

X=90/75

X=6/5

In Mathematics, Logarithm characteristics are utilised to solve Logarithm issues. Many algebraic characteristics, such as commutative, associative, and distributive, were taught to us in elementary school. There are five fundamental features of Logarithmic functions.

The Logarithmic number is connected with exponent and power, thus if xn = m, then logxm = n. As a result, we must also understand exponent law. The Logarithm of 10000 to base 10 is 4, for example, because 4 is the power to which ten must be raised to create 10000 : 104 = 10000, so log1010000 = 4.

We can represent the Logarithm of a product as a sum of Logarithms, the log of the quotient as a difference of logs, and the log of power as a product using these features.

Real number Logarithms are only seen in positive real numbers; negative and Complex numbers have Complex Logarithms.

### Logarithm Applications

Logarithms have a wide range of applications both within and outside of Mathematics. Let us look at a few examples of how Logarithms are used in everyday life:

• They are used to calculate the magnitude of an earthquake.
• Logarithms are used to calculate the amount of noise in decibels, such as the sound of a bell.
• Logarithms are used in Chemistry to determine acidity or pH level.
• They are used to calculate the growth of money at a given rate of interest.
• Logarithms are commonly used to calculate the time it takes for anything to decay or develop exponentially, such as bacteria growth or radioactive decay.
• They can also be utilised in computations that need multiplication to be converted to addition or vice versa.

Instead of a simple computation, we may utilise the Logarithm table to get the Logarithm of an integer. Before calculating the Logarithm of a number, we must first understand its characteristic and mantissa parts.

Characteristic Part – The characteristic component is the entire part of a number. Any number higher than one has a positive feature, and if it is one less than the number of digits to the left of the decimal point in a given integer, it has a negative characteristic. If the number is less than one, the characteristic is negative, and the number is one greater than the number of zeros to the right of the decimal point.

Mantissa Part – The mantissa portion is the decimal part of the Logarithm number, which should always be positive. If the mantissa part has a negative value, turn it into a positive value.

### How Do You Use a Log Table?

The process for determining the log value of a number using the log table is shown below. First, you must understand how to use the log table. The log table is provided as a resource for determining the values.

Step 1: Understand the Logarithm idea. Each log table may only be used with a certain basis. Log base 10 is the most often used form of Logarithm table.

Step 2: Determine the number’s characteristic and mantissa parts. To get the value of log1015.27, for example, first separate the characteristic and mantissa parts.

Part of Characteristic = 15

Part of the mantissa = 27

Step 3: Make use of a shared log table. Now, utilise row 15 to verify column 2 and write the matching value. As a result, the result is 1818.

Step 4: Calculate the mean difference using the Logarithm table. Slide your finger into the mean difference column 7 and row 15, and record the associated value as 20.

Step 5: Combine the values acquired in steps 3 and 4. That equals 1818 + 20 = 1838. As a result, the value 1838 represents the mantissa part.

Step 6: Locate the distinguishing feature. Because the number is between 10 and 100 (101 and 102), the distinguishing feature should be 1.

Step 7: Finally, combine the characteristic and mantissa parts to get 1.1838.

### Exemplification

Here is an example of utilising the Logarithm table to get the value of a Logarithmic function.

Determine the value of log102.872.

Solution:

Step 1: The characteristic component is 2 and the mantissa part is 872.

Step 2:  Examine rows 28 and 7 in the table. As a result, the resulting value is 4579.

Step 3: Examine the mean difference value for row 28 and the mean difference in column 2. The value associated with the row and column is 3.

Step 4: Adding the numbers from steps 2 and 3, we get 4582. This is the mantissa section.

Step 5: Because the number of digits to the left of the decimal part is one, the characteristic part is less than one. As a result, the characteristic portion is 0

Step 6: Finally, join the characteristic and mantissa parts. As a result, it becomes 0.4582.

As a result, log 2.872 = 0.4582

### Quiz Time

1. Which of the Following Statements is Not True?

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 addiction 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

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.

3. What are the applications of Logarithms?

Logarithm Applications

Logarithms have a wide range of applications both within and outside of Mathematics. Let us look at a few examples of how Logarithms are used in everyday life:

• They are used to calculate the magnitude of an earthquake.
• Logarithms are used to calculate the amount of noise in decibels, such as the sound of a bell.
• Logarithms are used in Chemistry to determine acidity or pH level.
• They are used to calculate the growth of money at a certain rate of interest.
• Logarithms are commonly used to calculate the time it takes for anything to decay or develop exponentially, such as bacteria growth or radioactive decay.
• They can also be utilised in computations that need multiplication to be converted to addition or vice versa.

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Exponents

### Log Formulas

Before learning log formulas, let us recall what are logs (logarithms). A logarithm is just another way of writing exponents. When we cannot solve a problem using the exponents, then we use logarithms. There are different log formulas that are derived by using the laws of exponents. Let us learn them using a few solved examples.

### What Are Log Formulas?

Before going to learn the log formulas, let us recall a few things. There are two types of logarithms, common logarithm (which is written as “log” and its base is 10 if not mentioned) and natural logarithm (which is written as “ln” and its base is always “e”). The below log formulas are shown for common logarithms. However, they are all applicable for natural logarithms as well. Here are the log formulas.

Here is the derivation of some important log formulas. We use the laws of exponents in the derivation of log formulas.

### Product Rule of logarithms

The product rule of logs is, logb (xy) = logb x + logb

y.

Derivation:

Let us assume that logb x = m and logb

y = n. Then by the definition of logarithm,

x = bm and y = bn.

Then xy = bm × bn = bm + n (by a law of exponents, a× an = am + n)

Converting xy = bm + n into logarithmic form, we get

m + n = logb

xy

Substituting the values logb x = m and logb

y = n here,

logb (xy) = logb x + logb y

### Quotient Rule of logarithms

The quotient rule of logs is, logb (x/y) = logb x – logb

y.

Derivation:

Let us assume that logb x = m and logb

y = n. Then by the definition of logarithm,

x = bm and y = bn.

Then x/y = bm / bn = bm – n (by a law of exponents, a/ an = am – n)

Converting x/y = bm – n into logarithmic form, we get

m – n = logb

(x/y)

Substituting the values logb x = m and logb

y = n here,

logb (x/y) = logb x – logb y

### Power Rule of Logarithms

The power rule of logarithms says logb ax = x logb

a.

Derivation:

Let logb

a = m. Then by the definition of logarithm, a = bm.

Raising both sides by x, we get

ax = (bm)x

ax = bmx (by a law of exponents, (am)n = amn)

Converting this back into logarithmic form,

logb

ax = m x

Substitute m = logb

a here,

logb ax = x logb a

### Change of Base Rule of Logarithms

The change of base rule of logs says logb a = (logc a) / (logc

b).

Derivation:

Assume that  logb a = x,  logc a = y, and  logc

b = z.

Converting these into exponential forms,

a = bx … (1)

a = cy … (2)

b = cz … (3)

From (1) and (2),

bx = cy

(cz)x = cy (from (3))

czx = cy

Since the bases are same, the powers also should be the same.

zx = y (or) x = y / z.

Substituting the values of x, y, and z here back,

logb a = (logc a) / (logc b).

We can see the applications of the log formulas in the section below.

### Examples Using Log Formulas

Example 1: Convert the following from exponential form to logarithmic form using the log formulas. a) 53 = 125  b) 3-3 = 1 / 27.

Solution:

Using the definition of the logarithm,

bx = a ⇒ logb

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: Compress the following expression as a single logarithm by using rules/properties of exponents. 5 log x + log y – 8 log z.

Solution:

To find: The compressed form of the given expression as a single logarithm using log 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 xa)

= 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 (∵ logb (x / y) = logb x – logb

y)

= 0 – log3 32 (∵ logb

1 = 0)

= – 2 log3 3 (∵ logb ax = x logb

a)

= -2 (1) (∵ logb

b = 1)

= -2

## FAQs on Log Formulas

### What Are Log Formulas?

The log formulas are related to logarithms and are very helpful while solving the problems of logarithms. Some important log formulas are:

• logb

(xy) = logb x + logb y logb (x / y) = logb x – logb y logb ax = x logb a logb a = (logc a) / (logc b)

### How To Derive Log Formulas?

The laws of exponents are used to derive the log formulas. We also use the definition of logarithm while deriving the log formulas. i.e. we convert the logarithmic form into exponential form and vice versa in the derivation. For a detailed derivation of log formulas, you can refer to the “What Are Log Formulas?” section of this page.

### What Are the Applications of Log Formulas?

The problems that cannot be solved using the exponents’ properties can be solved using logs. The log formulas are used to either compress a group of logarithms into a single logarithm or vice versa.

### What Is the Use of the Change of Base Formula (One of the Log Formulas)?

Here is an important use of the change of base formula. Usually, the calculators have options to calculate the logarithms of numbers with base 10 and with base “e”. To find the logarithms of numbers with other bases than 10 and “e”, we use the change of base formula. For example log2 3 = (log 3) / (log 2).

Some other very important formula are:

Suppose a, b , m, n are variables with positive integers and p as a real number. Then we have,

### The change of base rule

We can change the base of any logarithm by using the following rule:

Notes:

• When using this property, you can choose to change the logarithm to any base x\greenE xxstart color #0d923f, x, end color #0d923f.
• As always, the arguments of the logarithms must be positive and the bases of the logarithms must be positive and not equal to 1111 in order for this property to hold!

### Justifying the change of base rule

At this point, you might be thinking, “Great, but why does this rule work?”

### Properties of logarithms

Logarithms were quickly adopted by scientists because of various useful properties that simplified long, tedious calculations. In particular, scientists could find the product of two numbers m and n by looking up each number’s logarithm in a special table, adding the logarithms together, and then consulting the table again to find the number with that calculated logarithm (known as its antilogarithm). Expressed in terms of common logarithms, this relationship is given by log mn = log m + log n. For example, 100 × 1,000 can be calculated by looking up the logarithms of 100 (2) and 1,000 (3), adding the logarithms together (5), and then finding its antilogarithm (100,000) in the table. Similarly, division problems are converted into subtraction problems with logarithms: log m/n = log m − log n. This is not all; the calculation of powers and roots can be simplified with the use of logarithms. Logarithms can also be converted between any positive bases (except that 1 cannot be used as the base since all of its powers are equal to 1), as shown in the

table of logarithmic laws.

Only logarithms for numbers between 0 and 10 were typically included in logarithm tables. To obtain the logarithm of some number outside of this range, the number was first written in scientific notation as the product of its significant digits and its exponential power—for example, 358 would be written as 3.58 × 102, and 0.0046 would be written as 4.6 × 10−3. Then the logarithm of the significant digits—a decimal fraction between 0 and 1, known as the mantissa—would be found in a table. For example, to find the logarithm of 358, one would look up log 3.58 ≅ 0.55388. Therefore, log 358 = log 3.58 + log 100 = 0.55388 + 2 = 2.55388. In the example of a number with a negative exponent, such as 0.0046, one would look up log 4.6 ≅ 0.66276. Therefore, log 0.0046 = log 4.6 + log 0.001 = 0.66276 − 3 = −2.33724.

### History of logarithms

The invention of logarithms was foreshadowed by the comparison of arithmetic and geometric sequences. In a geometric sequence each term forms a constant ratio with its successor; for example, …1/1,000, 1/100, 1/10, 1, 10, 100, 1,000… has a common ratio of 10. In an arithmetic sequence each successive term differs by a constant, known as the common difference; for example, …−3, −2, −1, 0, 1, 2, 3… has a common difference of 1. Note that a geometric sequence can be written in terms of its common ratio; for the example geometric sequence given above: …10−3, 10−2, 10−1, 100, 101, 102, 103…. Multiplying two numbers in the geometric sequence, say 1/10 and 100, is equal to adding the corresponding exponents of the common ratio, −1 and 2, to obtain 101 = 10. Thus, multiplication is transformed into addition. The original comparison between the two series, however, was not based on any explicit use of the exponential notation; this was a later development. In 1620 the first table based on the concept of relating geometric and arithmetic sequences was published in Prague by the Swiss mathematician Joost Bürgi.

The Scottish mathematician John Napier published his discovery of logarithms in 1614. His purpose was to assist in the multiplication of quantities that were then called sines. The whole sine was the value of the side of a right-angled triangle with a large hypotenuse. (Napier’s original hypotenuse was 107.) His definition was given in terms of relative rates.

The logarithme, therefore, of any sine is a number very neerely expressing the line which increased equally in the meene time whiles the line of the whole sine decreased proportionally into that sine, both motions being equal timed and the beginning equally shift.

In cooperation with the English mathematician Henry Briggs, Napier adjusted his logarithm into its modern form. For the Naperian logarithm the comparison would be between points moving on a graduated straight line, the L point (for the logarithm) moving uniformly from minus infinity to plus infinity, the X point (for the sine) moving from zero to infinity at a speed proportional to its distance from zero. Furthermore, L is zero when X is one and their speed is equal at this point. The essence of Napier’s discovery is that this constitutes a generalization of the relation between the arithmetic and geometric series; i.e., multiplication and raising to a power of the values of the X point correspond to addition and multiplication of the values of the L point, respectively. In practice it is convenient to limit the L and X motion by the requirement that L = 1 at X = 10 in addition to the condition that X = 1 at L = 0. This change produced the Briggsian, or common, logarithm.

Napier died in 1617 and Briggs continued alone, publishing in 1624 a table of logarithms calculated to 14 decimal places for numbers from 1 to 20,000 and from 90,000 to 100,000. In 1628 the Dutch publisher Adriaan Vlacq brought out a 10-place table for values from 1 to 100,000, adding the missing 70,000 values. Both Briggs and Vlacq engaged in setting up log trigonometric tables. Such early tables were either to one-hundredth of a degree or to one minute of arc. In the 18th century, tables were published for 10-second intervals, which were convenient for seven-decimal-place tables. In general, finer intervals are required for calculating logarithmic functions of smaller numbers—for example, in the calculation of the functions log sin x and log tan x.

The availability of logarithms greatly influenced the form of plane and spherical trigonometry. The procedures of trigonometry were recast to produce formulas in which the operations that depend on logarithms are done all at once. The recourse to the tables then consisted of only two steps, obtaining logarithms and, after performing computations with the logarithms, obtaining antilogarithms.

## Logarithms

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