Logarithms

  • A Recap of Laws of Indices

Consider the following expression:

{2 \times 2 \times 2 \times 2 \times 2 =32}

A shorter version of this is 2^5. We say that 2 raised to 5 is 32. By this, we mean multiplying the number 2 by itself 5 times. The number 5 is termed as the index.

Whenever the index is a negative integer, i.e. 2^ {-3}, we actually mean

{2^{-3} = \frac {1}{2^3}}

Whenever the index is a reciprocal of any integer, i.e. 25^{\frac 1 2}, we actually mean

{25^{\frac 1 2} = \sqrt {25} = 5, \ i.e. \ 5^2 = 25}

I) For any number a except 0, a^0=1

II) For any number a, a^m \times a^n = a^{m+n}

III) For any number a, \frac {a^m}{a^n} = a^{m-n}

IV) For any number a, (a^m)^n = a^{m \times n} = (a^{n})^{m}

V) For any number a, ^{\frac m n} = \sqrt [n] {a^m}

  • Logarithms

Logarithms are just another way of representing a number raised to an index. Let m = a^x and a>0, a \ne 1, then

{x = log_{a} \ m}

This is known as the logarithmic form and the first representation is known as the indicial form.

x is known as the logarithm (short form log) and a is the base.

  • Important Points

I) Logarithms of negative numbers and zero are NOT defined.

II) Logarithm of 1 to any valid base is always 0.

III) Logarithm of any number to the same (valid) base is 1.

IV) This property is interesting:

{c = log_{a} \ a^c}

It proves that the logarithmic function and the exponential function are inverses of each other.

V) If a>1 and m>n, then

{log_a m > log_a  n}

VI) If {log_a m = log_a n}, then m=n.

  • Laws of Logarithms

If a,m,n are positive real numbers and a \ne 1, then

{log_a (m \times n) = log_a (m) \ + \ log_a (n)}

{log_a \frac {m}{n} = log_a (m) \ - \ log_a (n)}

{log_a (m^n) = n \times log_a (m)}

Let b be any real number such that b \ne 1, then

{log_a \ m = \frac {log_b m}{log_b a}}

  • Logs and Antilogs

We know that a^x =m gives x = log_a m. We write this alternatively as m equal to antilog of x to the base a.

  • 2 Important Bases, 10 and e

Commonly used bases of logarithms are 10 and e. The logarithms to base 10 are generally written as log and those to base e are generally written as ln.

Note: e is an irrational number, its approximate value is 2.71828.

  • Reading a Logarithmic Table

Standard log tables are available, with base 10. The log tables are handy, when it comes to multiplication, division, powers, roots of numbers. However, today, they’ve almost been replaced by electronic calculators.

Log of a number: The log contains 2 parts, the characteristic, the integral part and the mantissa, the decimal part.

I) The characteristic of any number is the power of 10, closest to the number, and smaller than that number. e.g., characteristic of 432 will be 2, because 100 = 10^2 is the power of 10, which is closest to 432 and is smaller than 432.

432 can be written as 4.32 \times 10^2. Hence, the characteristic is 2. Similarly, characteristic of 5 will be 0; 5 = 5.0 \times 10^0. Characteristic of 0.261 will be \bar 1; it can be written as 2.61 \times 10^{-1}.

Thus, if a number is written in a form, such that there is a single digit non-zero number to the left of decimal point, the power of 10 will be its characteristic.

II) The mantissa of a number is to be found out from log table. Irrespective of the position of decimal point, we look for the 1st non-zero digit of the number. The first 2 digits form a number. We look for a row, containing this number. The column number is the 3rd digit. The 4 digit number sharing the row and the column is the mantissa.

If the number contains more than 3 digits, the mean differences are subtracted/added as per the instructions given in the log table. Note that \textbf {mantissa is always positive}.

III) The logarithm is characteristic + mantissa.

{multiplication \equiv addition \ of \ logs}

{division \equiv subtraction \ of \ logs}

{(n)^{th} \ power \equiv multiplying \ log \ by \ (n)}

IV) The last step is taking antilogs. We make sure that the decimal part of the number so obtained is positive. Now, we look in antilog table and search for the rows and columns corresponding to the mantissa. Procedure is similar. (1st 2 digits – Row, 3rd digit – Column, 4th digit – Mean Differences)

V) The decimal point is put at a place, such that the final number has characteristic equal to the integral part of the logarithm.

Example:

{log (4235) = log (4.235 \times 10^3)}

Characteristic {=3} and mantissa {= 0.6268}.

Hence, {log (4235) = 3.6268}

{antilog (3.6268) = 4231.0}

{4231} is closer to {4235}. So, logarithms give us approximate answers. The error is {\frac {4235-4231}{4235} \times 100 \% = 0.0944 \%}, which is acceptable.

For {\sqrt {4235}}, we will divide the log of {4235} by {2} and then take its antilog, as {log_b a^m = m \times log_b a}.

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