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Common Logarithm

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Logarithm of a number to a given base is the index to which the base must be raised to produce the number. If there are three quantities indicated by a, x and n
then, a$^{x}$ = n.

Then x is said to be the logarithm of the number n to the base a symbolically it can be expressed as
log$_{a}$n = x
i.e, The logarithm of n to the base a is x.
Example : 10$^{y}$ = 1000
log$_{10}$ 1000 = 3
log of 1000 to the base 10 is 3.
If there is no base explicitly then it is said to be a common logarithm. Common logarithm has a base of 10 and it is easier to find common logs of powers of 10.

Definition

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A logarithm to the base 10 is known as common logarithm. Usually written as log or log$_{10}$. It is also known as the decadic logarithm or decimal logarithm. Common logarithms are very much useful for computations so mathematicians simply wrote log(x) instead of log$_{10}$(x).
Identity used to find the numerical value for logarithm to the base 10 is : 
log$_{10}$(x) = $\frac{log_{2}(x)}{log_{2}(10)}$

Formula

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If the logarithm of a number to any base is given, then the logarithm of the same number to any other base can be determined from the following relation
log$_{n}$m = $\frac{log_{a}m}{log_{a}n}$

Proof:
Let y = log$_{n}$m
Apply as exponents on n
we get n$^{y}$ = n$^{log_{n}m}$

n$^{y}$ = m

Take log$_{a}$ on both sides of the equation

log$_{a}n^{y}$ = log$_{a}$m

y log$_{a}$n = log$_{a}$m

Substitute the value of y

log$_{n}$m * log$_{a}$n = log$_{a}$m

Divide each side of the equation by log$_{a}$n we get

log$_{n}$m = $\frac{log_{a}m}{log_{a}n}$

Which is also known as change of base formula.
Following rules are necessary to understand logarithms.
1) log 1 = 0

2) log 10 = 1

3) log m$^{n}$ = n log(m)

4) log 10$^{n}$ = n

5) log(m . n) = log(m) + log(n)

6) log($\frac{m}{n}$) = log (m) - log(n)

7) $\sqrt[n]{m}$ = $\frac{1}{n}$ log m

Common Logarithm Vs Natural Logarithm

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Common Logarithms:
Logarithm of a number to the base 10 is known as common logarithm.
This system was first introduced by Henry Briggs. The base 10 in common logarithm is usually omitted and is useful for numerical calculations. We can use calculators to evaluate common logarithms and we use log button.

Natural logarithms:
Natural logarithms has a base e and is written as ln. e is an irrational and transcendental constant equal to 2.7183. For calculations we use "ln" button in the calculator.

For a real valued function of a real variable natural logarithm is defined as
e$^{ln(x)}$ = x when x > 0
$\Rightarrow$ ln(e$^{x}$) = x It is also known as Napierian logarithm after the name of John Napier.
The  number e is an incommensurable number and is equal to the infinite series.
1 + $\frac{1}{1!}$$\frac{1}{1!}$ + $\frac{1}{1!}$ + .............. $\infty$
Represented as ln x.
The function e$^{x}$ is called the exponential function.
Inverse of the exponential function is the natural logarithm, or logarithm with base e.

Problems

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Some of the problems based on common logarithms are given below:
Example 1: Find the value of x
log x = 3.2

Solution: Using the definition of logarithms the above given expression can be written as

x = 10$^{3.2}$
x = 1584.89

Therefore the value of x for the given problem is 1584.89

Example 2: What is log$_{3}$ 81 $^{4}$

Solution:
= 4 log$ _{3}$ 81

= 4 log$_{3}$ 3$^{4}$

= 16 log$_{3}$ 3

= 16

Example 3:  $\frac{log_{3}8}{log_{9}16 log_{4}10}$ = 3 log$_{10}$2

Solution:  Change all the log on LHS to the base 10 by using the formula

log$_{b}$x = $\frac{log_{a}x}{log_{a}b}$

log$_{3}$8 = $\frac{log_{10}8}{log_{10}3}$

= $\frac{log_{10}2^{3}}{log_{10}3}$

= $\frac{3log_{10}2}{log_{10}3}$

log$_{9}$16 = $\frac{log_{10}16}{log_{10}9}$

= $\frac{log_{10}2^{4}}{log_{10}3^{2}}$

= $\frac{4log_{10}2}{2log_{10}3}$

log$_{4}$10 = $\frac{log_{10}10}{log_{10}4}$

= $\frac{1}{log_{10}2^{2}}$

= $\frac{1}{2log_{10}2}$

LHS = $\frac{3log_{10}2}{log_{10}3}$ * $\frac{2log_{10}3}{4 log_{10}2}$ * $\frac{2 log_{10}2}{1}$

= 3log$_{10}$2

Example 4: Solve log 15 + log 35 - log 70 - log 8

Solution:
log(15 * 35) - log 70 - log 8

log 525 - log 70 - log 8

log($\frac{525}{70}$)- log 8

log (7.5) -  log8

= log($\frac{7.5}{8}$)

= -0.028

Common Logarithm Table

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Below is an image for common logarithm table:
Common Logarithm Table