Logarithms made easy : Complete Guide

Calculators are so standard these days for any mathematical calculation. Calculator applications in mobile phones have made them widely available and are easier to use. You do not have to carry your calculator everywhere. The most complex calculations are solved with a click.

However, earlier, there were no calculators back in the sixteenth century. All calculations had to be done manually, which was tedious and difficult and inaccurate. But later on, things changed. In 1550, a Scottish mathematician and theological writer, John Napier, invented a remarkable tool known as logarithms. With the help of this tool, many complex calculations were made easier to perform by the end of the century.

On August 1, 1550, this Scottish mathematician was born to Sir Archibald Napier and Janet Bothwell in Merchiston Castle, Edinburg. He also invented Napier’s bones and made usage of decimals as common in mathematics. 

Logarithms simplify multiplication calculation, which is primarily needed in astronomy. The word logarithm is derived from the Greek word “logos,” which means ratio or proportion and “arithmos”, which means number, which together makes ratio number.

Let us try to understand logarithms. Before we start with logarithms, we need to go through indices or exponents. Thorough knowledge of exponents is necessary to understand logarithms because logarithms and indices or exponents are very closely related. But first, let’s know the log values. 

Value of Log 1 to 10 

The Value of Log 1 to 10  are:

Log 1
Log 2 0.3010
Log 3 0.4771
Log 4 0.6020
Log 5 0.6989
Log 6 0.7781
Log 7 0.8450
Log 8 0.9030
Log 9 0.9542
Log 10 1

Introduction to Logarithms

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Look at the expression:

8 = 23

Here, number 3 is the power, or index or exponent, and 2 is the base. The above expression 8 = 23 can be written as log₂ 8 = 3. This is read as log to base 2 of 8 equals 3. Here we see logarithm is the same as power or exponent.

The two expressions:

8 = 23 log₂ 8 = 3

are both equal statements. The number of times we need to multiply 2 to get 8 is 3, so the logarithm is three.

The logarithm is how many times one number needs to be multiplied to get the other number. In the above expression 2 is multiplied three times to get the number 8. We deal with three numbers here:

  1. The base; the number we are multiplying ( in the above expression, it is 2)
  2. How many times it is multiplied (3, which is also the logarithm)
  3. The resultant (8)

Exponents 

Exponents and logarithms are closely related. 

An exponent says how many times a number is used in a multiplication. In the above expression,

23 = 2 x 2 x 2 = 8

so, here 2 is used 3 times in multiplication to get 8. Logarithm tells us what the exponent is; the above expression base is 2, and the exponent is 3. Hence, logarithm answers what the exponent we need is:

Example: What is log₆ 216 =?

63= 216 

So, an exponent 3 is needed to make 6 into 216 

Standard bases

Two bases are more widely used than the others. They are base 10 and base e.

Base 10 is sometimes written without the base as log (100), which means the base is 10. So any expression without a base can be assumed as 10. It is called a common logarithm. On the calculator, it is the “log” button.

There are two bases that are more widely used than the others. They are base 10 and base e.

Base 10 is sometimes written without the base as log (100), which means the base is 10. So any expression without a base can be assumed as 10. It is called a common logarithm. On the calculator, it is the “log” button.

Example: log (100) = log₁₀ (100) = 2 

Base e, the symbol e, is known as exponential constant. It is also known as Euler’s number. It has a value approximately equal to 2.718. This is called a natural logarithm. On the calculator, it is the “In” button.

Decimal values in logarithms

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In all the above examples, we have used whole numbers like 2 or 3, but they can even have decimal values like 2.5, 3.69, etc.

Example: What is log₁₀ (25)…

Take your calculator, click 25 and then click log

The result is 1.3979….

which means 101.3979… = 25 

10 with an exponent of 1.3979 equals 25 

Negative logarithms

What are negative logarithms? As logarithms are based on multiplication, the opposite of multiplication is division. So, a negative logarithm means how many times should we divide by the number.

Example: What is log₈ (0.125)?

which means 1 ÷ 8 = 0.125

Hence, log₈ (0.125) = -1

The first law of logarithm

The first law of logarithm states that if we want to multiply two numbers and find the logarithm of the result, it can be done by adding the logarithm of the two numbers.

logₐ xy = logₐ x logₐ y

Suppose, logₐ x = m and logₐ y = n 

Using the rule of indices, xy = and am= an m,

Hence, the logarithmic form of the statement (substitute the value of n and m)

logₐ xy = logₐ x logₐ y

The second law of logarithm

According to the second law of logarithm, finding the logarithm of power of a number can be done by multiplying the logarithm of the number by that power.

logₐxm = m logₐ x

Suppose, x = an i.e. logₐ x = an

raise both sides to power m 

xm= (am)n

Using rule of indices, xm= anm

The logarithmic form is logₐxm = nm = m logₐx

Thus, logₐxm =m logₐx

The third law of logarithm

The third law of logarithm states 

logₐ x/y = logₐ x − logₐ y

Suppose, logₐ x = m and logₐ y = n 

Consider x ÷ y

Using rule of indices, x/y = an ÷ am = an-m

In logarithmic form, logₐ x/y = logₐ x − logₐ y

The logarithmic of 1

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We all know any number raised to power zero is 1, i.e. a0= 1. The logarithmic form for this is logₐ 1 = 0

The logarithm of 1 to any base is 0.

Conclusion

These laws are used to solve logarithmic equations where the power is unknown. Logarithms are useful in solving exponential equations such as sound (measurement of a decibel), earthquake (Richter scale), measurement of pH balance, acidity and alkalinity. It is essential to do a lot of practice to master the techniques. If you aspire to be an engineer or work on computers, it is crucial to understand and learn logarithms well. 

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